CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> topological vector bundle (changes) in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> topological vector bundle (changes) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/7791/#Item_20" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #47 to #48: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/point-free+topology'>point-free topology</a>)</p> <p>see also <em><a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></em></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighbourhood base</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/diff/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/diff/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href='Top#UniversalConstructions'>Universal constructions</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>,</p> </li> <li> <p>fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+contractible+space'>locally contractible space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topological+space'>locally connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/semi-locally+simply-connected+topological+space'>locally simply-connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Scott+topology'>Scott topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/diff/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/diff/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology+of+uniform+convergence'>topology of uniform convergence</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sorgenfrey+line'>Sorgenfrey line</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hawaiian+earring+space'>Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings'>injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jordan+curve+theorem'>Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></p> </li> </ul> </div> <h4 id='bundles'>Bundles</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/bundle'>bundles</a></strong></p> <ul> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/parameterized+homotopy+theory'>stable</a>) <a class='existingWikiWord' href='/nlab/show/diff/parameterized+homotopy+theory'>parameterized homotopy theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+bundles+in+physics'>fiber bundles in physics</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory'>Chern-Weil theory</a></p> </li> </ul> <h2 id='sidebar_context'>Context</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/over-topos'>slice topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-topos'>slice (∞,1)-topos</a></p> </li> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/dependent+linear+type+theory'>linear</a>) <a class='existingWikiWord' href='/nlab/show/diff/dependent+type+theory'>dependent type theory</a></p> </li> </ul> <h2 id='sidebar_classes_of_bundles'>Classes of bundles</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/retractive+space'>retractive space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>fiber bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/fiber+infinity-bundle'>fiber ∞-bundle</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/numerable+fiber+bundle'>numerable bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere+fiber+bundle'>sphere bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+bundle'>projective bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+bundle'>principal bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/principal+infinity-bundle'>principal ∞-bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+2-bundle'>principal 2-bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/principal+3-bundle'>principal 3-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle+bundle'>circle bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/circle+n-bundle+with+connection'>circle n-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/double+cover'>orientation bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spinor+bundle'>spinor bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/stringor+bundle'>stringor bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/gerbe'>gerbe</a>, <a class='existingWikiWord' href='/nlab/show/diff/2-gerbe'>2-gerbe</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-gerbe'>∞-gerbe</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/local+coefficient+bundle'>local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/2-vector+bundle'>2-vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-vector bundle</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/real+vector+bundle'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+vector+bundle'>complex</a>/<a class='existingWikiWord' href='/nlab/show/diff/holomorphic+vector+bundle'>holomorphic</a>, <a class='existingWikiWord' href='/nlab/show/diff/quaternionic+vector+bundle'>quaternionic</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological</a>, <a class='existingWikiWord' href='/nlab/show/diff/differentiable+vector+bundle'>differentiable</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+vector+bundle'>algebraic</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+vector+bundle'>with connection</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>line bundle</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/holomorphic+vector+bundle'>holomorphic</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+line+bundle'>algebraic</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/cubical+structure+on+a+line+bundle'>cubical structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory'>Chern-Weil theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tensor+category'>tensor category</a> <a class='existingWikiWord' href='/nlab/show/diff/Vect%28X%29'>of vector bundles</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/VectBund'>VectBund</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/direct+sum+of+vector+bundles'>direct sum</a>, <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+vector+bundles'>tensor product</a>, <a class='existingWikiWord' href='/nlab/show/diff/external+tensor+product+of+vector+bundles'>external tensor product</a>, <a class='existingWikiWord' href='/nlab/show/diff/inner+product+on+vector+bundles'>inner product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dual+vector+bundle'>dual vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+vector+bundle'>stable vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/virtual+vector+bundle'>virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/parametrized+spectrum'>bundle of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/natural+bundle'>natural bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivariant+bundle'>equivariant bundle</a></p> </li> </ul> <h2 id='sidebar_universal_bundles'>Universal bundles</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+principal+bundle'>universal principal bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/universal+principal+infinity-bundle'>universal principal ∞-bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+vector+bundle'>universal vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/universal+complex+line+bundle'>universal complex line bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subobject+classifier'>subobject classifier</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos'>object classifier</a></p> </li> </ul> <h2 id='sidebar_presentations'>Presentations</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bundle+gerbe'>bundle gerbe</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/groupal+model+for+universal+principal+infinity-bundles'>groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/microbundle'>microbundle</a></p> </li> </ul> <h2 id='sidebar_examples'>Examples</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+bundle'>empty bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/zero+bundle'>zero bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+bundle'>normal bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tautological+line+bundle'>tautological line bundle</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/basic+complex+line+bundle+on+the+2-sphere'>basic line bundle on the 2-sphere</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hopf+fibration'>Hopf fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/canonical+bundle'>canonical line bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/prequantum+line+bundle'>prequantum circle bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/prequantum+circle+n-bundle'>prequantum circle n-bundle</a></p> </li> </ul> <h2 id='sidebar_constructions'>Constructions</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/clutching+construction'>clutching construction</a></li> </ul> </div> <h4 id='linear_algebra'>Linear algebra</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a></strong></p> <p>flavors: <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/diff/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/diff/cohesive+homotopy+theory'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/diff/directed+homotopy+theory'>directed</a>…</p> <p>models: <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/diff/localic+homotopy+theory'>localic</a>, …</p> <p>see also <strong><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>higher homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/spherical+object'>spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/diff/cofibration+category'>cofibration category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Waldhausen+category'>Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+universal+bundle'>universal bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+at+geometric+homotopies'>homotopy localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+interval+object'>infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+a+topos'>fundamental group of a topos</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+category'>fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Galois+theory'>Galois theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a><ul><li><a href='#InTermsOfSliceCategories'>In terms of slice categories</a></li><li><a href='#in_components'>In components</a></li><li><a href='#TransitionFunctionsAndCechCohomology'>Transition functions and Cech cohomology</a></li></ul></li><li><a href='#Examples'>Examples</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#BasicProperties'>Basic properties</a></li><li><a href='#DirectSummandBundles'>Direct summand bundles</a></li><li><a href='#ConcordanceOfTopolgicslVectorBundles'>Concordance</a></li><li><a href='#OverClosedSubspaces'>Over closed subspaces</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A <em>topological vector bundle</em> is a <a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a> in the context of <em><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></em>: a continuously varying collection of <a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a> over a given <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>.</p> <p>For more survey and motivation see at <em><a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a></em>. Here we discuss the details of the general concept in <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a>. See also <em><a class='existingWikiWord' href='/nlab/show/diff/differentiable+vector+bundle'>differentiable vector bundle</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+vector+bundle'>algebraic vector bundle</a></em>.</p> <h2 id='definition'>Definition</h2> <p><span> We first give the more abstract definiton in terms of<del class='diffdel'> slice</del><del class='diffdel'> categories</del><del class='diffdel'> (def.</del></span><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/over+category'>slice categories</a></ins><ins class='diffins'> (def. </ins><a class='maruku-ref' href='#TopologicalVectorBundleInTermsOfSliceCategories'>1</a> below) and then unwind this to the traditional definition (def <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a> below).</p> <p>In the following</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> is either the <a class='existingWikiWord' href='/nlab/show/diff/topological+field'>topological field</a></p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>k = \mathbb{R}</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real numbers</a></p> </li> <li> <p>or <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>k = \mathbb{C}</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/complex+number'>complex numbers</a></p> </li> </ul> <p>equipped with the <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>.</p> </li> <li> <p><em><a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a></em> means <em><a class='existingWikiWord' href='/nlab/show/diff/finite-dimensional+vector+space'>finite dimensional vector space</a></em>.</p> </li> </ul> <h3 id='InTermsOfSliceCategories'>In terms of slice categories</h3> <div class='num_defn' id='TopologicalVectorBundleInTermsOfSliceCategories'> <h6 id='definition_2'>Definition</h6> <p><strong>(topological vector bundles in terms of slice categories)</strong></p> <p>Write <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> for the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>, and for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>X \in Top</annotation></semantics></math> a space, write <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>Top_{/X}</annotation></semantics></math> for its <a class='existingWikiWord' href='/nlab/show/diff/over+category'>slice category</a> over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. The <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a> in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>Top_{/X}</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>fiber product</a> over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math>, which we denote by <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><msub><mo>×</mo> <mi>X</mi></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-) \times_X (-)</annotation></semantics></math>. Observe <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>×</mo><mi>k</mi><mo>→</mo><mi>X</mi><mo stretchy='false'>]</mo><mo>∈</mo><msub><mi>Top</mi> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>[X \times k \to X] \in Top_{/X}</annotation></semantics></math> is canonically a <a class='existingWikiWord' href='/nlab/show/diff/field'>field</a> <a class='existingWikiWord' href='/nlab/show/diff/internalization'>internal</a> to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>Top_{/X}</annotation></semantics></math></p> <p>A <em>topological vector bundle</em> over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>X \in Top</annotation></semantics></math> is</p> <ol> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>E</mi><mover><mo>→</mo><mi>π</mi></mover><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[E \overset{\pi}{\to} X]</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>Top_{/X}</annotation></semantics></math></p> </li> <li> <p>with the <a class='existingWikiWord' href='/nlab/show/diff/structure'>structure</a> of an <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>X \times k</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a>-object <a class='existingWikiWord' href='/nlab/show/diff/internalization'>internal</a> to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>Top_{/X}</annotation></semantics></math>, hence</p> <ol> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>+</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>E</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'> (-)+(-) \;\colon\; E \times_X E \to E</annotation></semantics></math></p> </li> <li> <p>a morphism <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>⋅</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>k</mi><mo>×</mo><mi>E</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>(-)\cdot(-) \;\colon\; k \times E \to E</annotation></semantics></math></p> </li> </ol> <p><span> which satisfy the<del class='diffdel'> vector</del><del class='diffdel'> space</del><del class='diffdel'> axioms</del></span><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a></ins><ins class='diffins'> </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/axiom'>axioms</a></ins></p> </li> </ol> <p>such that</p> <ul> <li> <p>(<a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>local triviality</a>) there exists</p> <ol> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math>, regarded via the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>≔</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U \coloneqq \underset{i \in I}{\sqcup} U_i</annotation></semantics></math> of the patches as the object <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>U</mi><mo>→</mo><mi>X</mi><mo stretchy='false'>]</mo><mo>∈</mo><msub><mi>Top</mi> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> [U \to X] \in Top_{/X}</annotation></semantics></math>,</p> </li> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> of vector space objects in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><mo stretchy='false'>/</mo><mi>U</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>Top_{/U}</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><del class='diffmod'><mo>×</mo></del><ins class='diffmod'><msub><mo>×</mo> <mi>I</mi></msub></ins><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mo>≃</mo></mover><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>E</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'><span> U<del class='diffmod'> \times</del><ins class='diffmod'> \times_I</ins><del class='diffmod'> \mathbb{R}^n</del><ins class='diffmod'> \mathbb{R}^{n}</ins> \overset{\simeq}{\longrightarrow} U \times_X E \,,</span></annotation></semantics></math></div> <p><span> for<del class='diffdel'> some</del></span><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo><span><del class='diffmod'> ∈</del><ins class='diffmod'> :</ins></span></mo><ins class='diffins'><mi>I</mi></ins><ins class='diffins'><mo>→</mo></ins><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'><span> n<del class='diffmod'> \in</del><ins class='diffmod'> \colon</ins><ins class='diffins'> I</ins><ins class='diffins'> \to</ins> \mathbb{N}</span></annotation></semantics></math><span><del class='diffmod'> ,</del><ins class='diffmod'> </ins><del class='diffmod'> where</del><ins class='diffmod'> some</ins></span><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><del class='diffmod'><mo stretchy='false'>[</mo></del><ins class='diffmod'><mi>I</mi></ins><del class='diffdel'><mi>U</mi></del><del class='diffdel'><mo>×</mo></del><del class='diffdel'><msup><mi>k</mi> <mi>n</mi></msup></del><del class='diffdel'><mover><mo>→</mo><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mover></del><del class='diffdel'><mi>X</mi></del><del class='diffdel'><mo stretchy='false'>]</mo></del></mrow><annotation encoding='application/x-tex'><span><del class='diffmod'> [U</del><ins class='diffmod'> I</ins><del class='diffdel'> \times</del><del class='diffdel'> k^n</del><del class='diffdel'> \overset{pr_1}{\to}</del><del class='diffdel'> X]</del></span></annotation></semantics></math><span><del class='diffmod'> </del><ins class='diffmod'> -</ins><del class='diffdel'> and</del></span><del class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>E</mi><mover><mo>→</mo><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mover><mi>U</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[U \times_X E \overset{pr_1}{\to} U]</annotation></semantics></math></del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/family'>indexed set</a></ins><span> <del class='diffmod'> are</del><ins class='diffmod'> of</ins><del class='diffdel'> regarded</del><del class='diffdel'> as</del><del class='diffdel'> a</del><del class='diffdel'> vector</del><del class='diffdel'> space</del><del class='diffdel'> objects</del><del class='diffdel'> in</del></span><del class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>U</mi></msub></mrow><annotation encoding='application/x-tex'>Top_{U}</annotation></semantics></math></del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/natural+number'>natural numbers</a></ins><span><del class='diffmod'> </del><ins class='diffmod'> ,</ins><del class='diffdel'> in</del><del class='diffdel'> the</del><del class='diffdel'> canonical</del><del class='diffdel'> way.</del></span></p> </li> </ol><ins class='diffins'> </ins><ins class='diffins'><p>where <math class='maruku-mathml' display='inline' id='mathml_325f1d1863d15fc5acaa4072f520e8419a54b72e_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>U</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mover><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[U \times k^n \overset{pr_1}{\to} X]</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_325f1d1863d15fc5acaa4072f520e8419a54b72e_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>E</mi><mover><mo>→</mo><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mover><mi>U</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[U \times_X E \overset{pr_1}{\to} U]</annotation></semantics></math> are regarded as vector space objects in <math class='maruku-mathml' display='inline' id='mathml_325f1d1863d15fc5acaa4072f520e8419a54b72e_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>U</mi></msub></mrow><annotation encoding='application/x-tex'>Top_{U}</annotation></semantics></math> in the canonical way.</p></ins> </li> </ul> <p>It follows that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><del class='diffdel'><mo>∈</mo></del><del class='diffdel'><mi>ℕ</mi></del></mrow><annotation encoding='application/x-tex'><span> n<del class='diffdel'> \in</del><del class='diffdel'> \mathbb{N}</del></span></annotation></semantics></math> is constant on <a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected components</a><span><del class='diffmod'> .</del><ins class='diffmod'> </ins><del class='diffdel'> Often</del><del class='diffdel'> this</del><del class='diffdel'> is</del><del class='diffdel'> required</del><del class='diffdel'> to</del><del class='diffdel'> be</del><del class='diffdel'> constant</del><del class='diffdel'> on</del><del class='diffdel'> all</del> of</span><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math><ins class='diffins'>. Often this is required to be constant on all of </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_325f1d1863d15fc5acaa4072f520e8419a54b72e_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></ins> and then called the <em><a class='existingWikiWord' href='/nlab/show/diff/rank'>rank</a></em> of the vector bundle.</p> <p>A <em><a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a></em> of topological vector bundles is simple a homomorphism of vector space objects in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><mo stretchy='false'>/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>Top_{/X}</annotation></semantics></math>.</p> <p>Topological vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and homomorphisms between them constitutes a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>, usually denoted <a class='existingWikiWord' href='/nlab/show/diff/Vect%28X%29'>Vect(X)</a>.</p> </div> <p>Notice that viewed in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>, the last condition means that there is a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>U</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><mi>U</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow /></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd /> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>pb</mi><mo stretchy='false'>)</mo></mtd> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>π</mi></mpadded></msup></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>U</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ U \times k^n &\overset{\simeq}{\longrightarrow}& U \times_X E &\overset{}{\longrightarrow}& E \\ & \searrow & \downarrow &(pb)& \downarrow^{\mathrlap{\pi}} \\ && U &\longrightarrow& X } </annotation></semantics></math></div> <p>where the square is a <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback square</a> and the <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a> in the top left is fiber-wise linear.</p> <p>If we say this yet more explicitly, it yields the definition as found in the traditional textbooks:</p> <h3 id='in_components'>In components</h3> <div class='num_defn' id='TopologicalVectorBundle'> <h6 id='definition_3'>Definition</h6> <p><strong>(topological vector bundle in components)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>. Then a <em>topological vector bundle</em> over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is</p> <ol> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>;</p> </li> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>π</mi></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \overset{\pi}{\to} X</annotation></semantics></math></p> </li> <li> <p>for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> the stucture of a <a class='existingWikiWord' href='/nlab/show/diff/finite-dimensional+vector+space'>finite-dimensional</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a> on the <a class='existingWikiWord' href='/nlab/show/diff/preimage'>pre-image</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>x</mi></msub><mo>≔</mo><msup><mi>π</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>⊂</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>E_x \coloneqq \pi^{-1}(\{x\}) \subset E</annotation></semantics></math></div></li> </ol> <p>such that this is <a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>locally trivial</a> in that there exists</p> <ol> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math>,</p> </li> <li> <p>for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i \in I</annotation></semantics></math> an <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>n</mi> <mi>i</mi></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n_i \in \mathbb{N}</annotation></semantics></math> and a <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>π</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>⊂</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'> \phi_i \;\colon\; U_i \times k^{n_i} \overset{\simeq}{\longrightarrow} \pi^{-1}(U_i) \subset E </annotation></semantics></math></div> <p>from the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real numbers</a> (equipped with their <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>) to the restriction of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math>, such that</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\phi_i</annotation></semantics></math> is a map over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math> in that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>=</mo><msub><mi>pr</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\pi \circ \phi_i = pr_1</annotation></semantics></math>, hence in that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mo stretchy='false'>)</mo><mo>⊂</mo><msup><mi>π</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\phi_i(\{x\} \times k^{n_i}) \subset \pi^{-1}(\{x\})</annotation></semantics></math></p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\phi_i</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/linear+map'>linear map</a> in each fiber in that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∀</mo><mrow><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow></munder><mrow><mo>(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msup><mi>k</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup><mover><mo>⟶</mo><mtext>linear</mtext></mover><msub><mi>E</mi> <mi>x</mi></msub><mo>=</mo><msup><mi>π</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \underset{x \in U_i}{\forall} \left( \phi_i(x) \;\colon\; k^{n_i} \overset{\text{linear}}{\longrightarrow} E_x = \pi^{-1}(\{x\}) \right) \,. </annotation></semantics></math></div></li> </ol> </li> </ol> <p>Often, but not always, it is required that the numbers <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>n_i</annotation></semantics></math> are all equal to some <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>, for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i \in I</annotation></semantics></math>, hence that the vector space fibers all have the same <a class='existingWikiWord' href='/nlab/show/diff/dimension'>dimension</a>. In this case one says that the vector bundle has <em><a class='existingWikiWord' href='/nlab/show/diff/rank'>rank</a></em> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>. (Over a <a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected topological space</a> this is automatic, but the fiber dimension may be distinct over distinct <a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected components</a>.)</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>E</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow></mover><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[E_1 \overset{\pi_1}{\to} X]</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>E</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mi>π</mi> <mn>2</mn></msub></mrow></mover><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[E_2 \overset{\pi_2}{\to} X]</annotation></semantics></math> two topological vector bundles over the same base space, then a <em><a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a></em> between them is</p> <ul> <li>a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><msub><mi>E</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>f \colon E_1 \longrightarrow E_2</annotation></semantics></math></li> </ul> <p>such that</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> respects the <a class='existingWikiWord' href='/nlab/show/diff/projection'>projections</a>: <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>2</mn></msub><mo>∘</mo><mi>f</mi><mo>=</mo><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\pi_2 \circ f = \pi_1</annotation></semantics></math>;</p> </li> <li> <p>for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> we have that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><msub><mo stretchy='false'>|</mo> <mi>x</mi></msub><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>)</mo> <mi>x</mi></msub><mo>→</mo><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>)</mo> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>f|_x \colon (E_1)_x \to (E_2)_x</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/linear+map'>linear map</a>.</p> </li> </ol> </div> <div class='num_remark' id='TopologicalVectorBundlesCategory'> <h6 id='remark'>Remark</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/Vect%28X%29'>category of topological vector bundles</a>)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, there is the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> whose</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> are the topological vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> are the topological vector bundle homomorphisms</p> </li> </ul> <p>according to def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>. This category usually denoted <a class='existingWikiWord' href='/nlab/show/diff/Vect%28X%29'>Vect(X)</a>.</p> <p>We write <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Vect</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>Vect(X)_{/\sim}</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> of <a class='existingWikiWord' href='/nlab/show/diff/isomorphism+class'>isomorphism classes</a> of this category.</p> </div> <p>\begin{remark} \label{FiberwiseOperations} <strong>(fiberwise operations)</strong> \linebreak The <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/FinDimVect'>FinDimVect</a> of <a class='existingWikiWord' href='/nlab/show/diff/finite-dimensional+vector+space'>finite dimensional vector spaces</a> over a <a class='existingWikiWord' href='/nlab/show/diff/topological+field'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/ground+ring'>ground field</a> is canonically a <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched category</a>, and so are hence its <a class='existingWikiWord' href='/nlab/show/diff/product+category'>product categories</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>FinDimVect</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>FinDimVect^{n}</annotation></semantics></math>, for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>. Any <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+functor'>enriched functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msup><mi>FinDimVect</mi> <mi>n</mi></msup><mo>⟶</mo><mi>FinDimVect</mi></mrow><annotation encoding='application/x-tex'> F \;\colon\; FinDimVect^n \longrightarrow FinDimVect </annotation></semantics></math></div> <p>induces a functorial construction of new topological vector bundles <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo>^</mo></mover><mo stretchy='false'>(</mo><msub><mi>𝒱</mi> <mn>1</mn></msub><mo>,</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>𝒱</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\widehat{F}(\mathcal{V}_1,, \cdots, \mathcal{V}_n)</annotation></semantics></math> from any <a class='existingWikiWord' href='/nlab/show/diff/tuple'>n-tuple</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>𝒱</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>𝒱</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>𝒱</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{V}_1, \mathcal{V}_2 , \cdots, \mathcal{V}_n)</annotation></semantics></math> of vector bundles over the same base space <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, by taking the new <a class='existingWikiWord' href='/nlab/show/diff/fiber'>fiber</a> over a point <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>b \in B</annotation></semantics></math> to be (e.g. <a href='#MilnorStasheff74'>Milnor & Stasheff 1974, p. 32</a>):</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>𝒱</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><msub><mi>𝒱</mi> <mi>n</mi></msub><msub><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>b</mi></msub><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mi>F</mi><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><msub><mi>𝒱</mi> <mn>1</mn></msub><msub><mo stretchy='false'>)</mo> <mi>b</mi></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><mo stretchy='false'>(</mo><msub><mi>𝒱</mi> <mi>n</mi></msub><msub><mo stretchy='false'>)</mo> <mi>b</mi></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> F \big( \mathcal{V}_1, \cdots \mathcal{V}_n \big) _b \;\coloneqq\; F \big( (\mathcal{V}_1)_b, \cdots, (\mathcal{V}_n)_b \big) \,. </annotation></semantics></math></div> <p>For example:</p> <ul> <li> <p>if <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mspace width='thinmathspace' /><mo>≔</mo><mspace width='thinmathspace' /><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><mi>FinDimVect</mi><mo>⟶</mo><mi>FinDimVect</mi></mrow><annotation encoding='application/x-tex'>F \,\coloneqq\, (-)^\ast \,\colon\, FinDimVect \longrightarrow FinDimVect</annotation></semantics></math> is the operation of forming <a class='existingWikiWord' href='/nlab/show/diff/dual+vector+space'>dual vector spaces</a>, then <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo>^</mo></mover></mrow><annotation encoding='application/x-tex'>\widehat{F}</annotation></semantics></math> constructs the fiberwise <a class='existingWikiWord' href='/nlab/show/diff/dual+vector+bundle'>dual vector bundle</a>;</p> </li> <li> <p>if <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mspace width='thinmathspace' /><mo>≔</mo><mspace width='thinmathspace' /><mi>det</mi><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><mi>FinDimVect</mi><mo>⟶</mo><mi>FinDimVect</mi></mrow><annotation encoding='application/x-tex'>F \,\coloneqq\, det \,\colon\, FinDimVect \longrightarrow FinDimVect</annotation></semantics></math> is the operation of forming <a class='existingWikiWord' href='/nlab/show/diff/determinant'>determinants</a>, then <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo>^</mo></mover></mrow><annotation encoding='application/x-tex'>\widehat{F}</annotation></semantics></math> is the construction of fiberwise <a class='existingWikiWord' href='/nlab/show/diff/determinant+line+bundle'>determinant line bundles</a>;</p> </li> <li> <p>if <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mspace width='thinmathspace' /><mo>≔</mo><mspace width='thinmathspace' /><mo>⊕</mo><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><msup><mi>FinDimVect</mi> <mn>2</mn></msup><mo>⟶</mo><mi>FinDimVect</mi></mrow><annotation encoding='application/x-tex'>F \,\coloneqq\, \oplus \,\colon\, FinDimVect^2 \longrightarrow FinDimVect</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a> of vector space, then <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo>^</mo></mover></mrow><annotation encoding='application/x-tex'>\widehat{F}</annotation></semantics></math> constructs the fiberwise <a class='existingWikiWord' href='/nlab/show/diff/direct+sum+of+vector+bundles'>direct sum of vector bundles</a> (“<a class='existingWikiWord' href='/nlab/show/diff/direct+sum+of+vector+bundles'>Whitney sum</a>”);</p> </li> <li> <p>if <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mspace width='thinmathspace' /><mo>≔</mo><mspace width='thinmathspace' /><mo>⊗</mo><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><msup><mi>FinDimVect</mi> <mn>2</mn></msup><mo>⟶</mo><mi>FinDimVect</mi></mrow><annotation encoding='application/x-tex'>F \,\coloneqq\, \otimes \,\colon\, FinDimVect^2 \longrightarrow FinDimVect</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+vector+spaces'>tensor product of vector spaces</a>, then <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo>^</mo></mover></mrow><annotation encoding='application/x-tex'>\widehat{F}</annotation></semantics></math> constructs the fiberwise <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+vector+bundles'>tensor product of vector bundles</a>.</p> </li> </ul> <p>\end{remark}</p> <div class='num_remark' id='TerminologyVectorBundles'> <h6 id='remark_2'>Remark</h6> <p><strong>(some terminology)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> be as in def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>. Then:</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>k = \mathbb{R}</annotation></semantics></math> one speaks of <em><a class='existingWikiWord' href='/nlab/show/diff/real+vector+bundle'>real vector bundles</a></em>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>k = \mathbb{C}</annotation></semantics></math> one speaks of <em><a class='existingWikiWord' href='/nlab/show/diff/complex+vector+bundle'>complex vector bundles</a></em>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n = 1</annotation></semantics></math> one speaks of <em><a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>line bundles</a></em>, in particular of <em><a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>real line bundles</a></em> and of <em><a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>complex line bundles</a></em>.</p> </div> <div class='num_remark' id='CommonOpenCoverLocalTrivialization'> <h6 id='remark_3'>Remark</h6> <p><strong>(any two topologial vector bundles have <a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>local trivialization</a> over a common <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>E</mi> <mn>1</mn></msub><mo>→</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[E_1 \to X]</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo>→</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[E_2 \to X]</annotation></semantics></math> be two topological vector bundles (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>). Then there always exists an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> such that both bundles have a <a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>local trivialization</a> over this cover.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>By definition we may find two possibly different open covers <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mn>1</mn></msubsup><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow><mo>∈</mo><msub><mi>I</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U^1_{i_1} \subset X\}_{{i_1} \in I_1}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow> <mn>2</mn></msubsup><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><msub><mi>i</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>I</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U^2_{i_2} \subset X\}_{i_2 \in I_2}</annotation></semantics></math> with local tivializations <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mn>1</mn></msubsup><munderover><mo>→</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mn>1</mn></msubsup></mrow></munderover><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>|</mo> <mrow><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mn>1</mn></msubsup></mrow></msub><msub><mo stretchy='false'>}</mo> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>I</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\{ U^1_{i_1} \underoverset{\simeq}{\phi^1_{i_1}}{\to} E_1\vert_{U^1_{i_1}} \}_{i_1 \in I_1}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow> <mn>2</mn></msubsup><munderover><mo>→</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow> <mn>2</mn></msubsup></mrow></munderover><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mrow><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow> <mn>2</mn></msubsup></mrow></msub><msub><mo stretchy='false'>}</mo> <mrow><msub><mi>i</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>I</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\{ U^2_{i_2} \underoverset{\simeq}{\phi^2_{i_2}}{\to} E_2\vert_{U^2_{i_2}} \}_{i_2 \in I_2}</annotation></semantics></math>.</p> <p>The <em>joint <a class='existingWikiWord' href='/nlab/show/diff/refinement'>refinement</a></em> of these two covers is the open cover</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></msub><mo>≔</mo><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mn>1</mn></msubsup><mo>∩</mo><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow> <mn>2</mn></msubsup><mo>⊂</mo><mi>X</mi><mo>}</mo></mrow> <mrow><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>I</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>I</mi> <mn>2</mn></msub></mrow></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \left\{ U_{i_1, i_2} \coloneqq U^1_{i_1} \cap U^2_{i_2} \subset X \right\}_{(i_1, i_2) \in I_1 \times I_2} \,. </annotation></semantics></math></div> <p>The original local trivializations restrict to local trivializations on this finer cover</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></msub><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mn>1</mn></msubsup><msub><mo stretchy='false'>|</mo> <mrow><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow> <mn>2</mn></msubsup></mrow></msub></mrow></munderover><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></msub></mrow></msub><mo>}</mo></mrow> <mrow><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>I</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>I</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding='application/x-tex'> \left\{ U_{i_1, i_2} \underoverset{\simeq}{\phi^1_{i_1}\vert_{U^2_{i_2}}}{\longrightarrow} E_1\vert_{U_{i_1, i_2}} \right\}_{(i_1, i_2) \in I_1 \times I_2} </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></msub><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow> <mn>2</mn></msubsup><msub><mo stretchy='false'>|</mo> <mrow><msubsup><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow> <mn>1</mn></msubsup></mrow></msub></mrow></munderover><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub></mrow></msub></mrow></msub><mo>}</mo></mrow> <mrow><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>I</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>I</mi> <mn>2</mn></msub></mrow></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \left\{ U_{i_1, i_2} \underoverset{\simeq}{\phi^2_{i_2}\vert_{U^1_{i_1}}}{\longrightarrow} E_2\vert_{U_{i_1, i_2}} \right\}_{(i_1, i_2) \in I_1 \times I_2} \,. </annotation></semantics></math></div></div> <div class='num_example' id='TrivialTopologicalVectorBundle'> <h6 id='example'>Example</h6> <p><strong>(trivial topological vector bundle and (local) trivialization)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> any <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>, we have that the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'> X \times k^n \overset{pr_1}{\to} X </annotation></semantics></math></div> <p>canonically becomes a topological vector bundle over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>). This is called the <em><a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial vector bundle</a></em> of <a class='existingWikiWord' href='/nlab/show/diff/rank'>rank</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>Given any topological vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math>, then a choice of <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> to a trivial bundle (if it exists)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mover><mo>⟶</mo><mo>≃</mo></mover><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'> E \overset{\simeq}{\longrightarrow} X \times k^n </annotation></semantics></math></div> <p>is called a <em>trivialization</em> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>. A vector bundle for which a trivialization exists is called <em>trivializable</em>.</p> <p>Accordingly, the <a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>local triviality</a> condition in the definition of topological vector bundles (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>) says that they are locally isomorphic to the trivial vector bundle. One also says that the data consisting of an open cover <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> and the <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphisms</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>→</mo><mo>≃</mo></mover><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>}</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \left\{ U_i \times k^n \overset{\simeq}{\to} E|_{U_i} \right\}_{i \in I} </annotation></semantics></math></div> <p>as in def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a> constitute a <em><a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>local trivialization</a></em> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>.</p> </div> <div class='num_example' id='VectorBundleSections'> <h6 id='example_2'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/section'>section</a> of a <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>π</mi></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \overset{\pi}{\to} X</annotation></semantics></math> be a topological vector bundle (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>).</p> <p>Then a <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> of vector bundles from the <a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial</a> <a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>line bundle</a> (example <a class='maruku-ref' href='#TrivialTopologicalVectorBundle'>1</a>, remark <a class='maruku-ref' href='#TerminologyVectorBundles'>2</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>×</mo><mi>k</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'> f \;\colon\; X \times k \longrightarrow E </annotation></semantics></math></div> <p>is, by fiberwise linearity, equivalently a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'> \sigma \;\colon\; X \longrightarrow E </annotation></semantics></math></div> <p>such that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo>∘</mo><mi>σ</mi><mo>=</mo><msub><mi>id</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>\pi \circ \sigma = id_X</annotation></semantics></math>;</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>c</mi><mi>σ</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> f(x, c) = c \sigma(x) </annotation></semantics></math></div> <p>Such functions <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>\sigma \colon X \to E</annotation></semantics></math> are called <em><a class='existingWikiWord' href='/nlab/show/diff/section'>sections</a></em> (or <em>cross-sections</em>) of the vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>.</p> </div> <div class='num_example' id='TopologicalVetorSubbundle'> <h6 id='example_3'>Example</h6> <p><strong>(topological vector sub-bundle)</strong></p> <p>Given a topological vector bundel <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>), then a <em>sub-bundle</em> is a homomorphism of topological vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><mo>′</mo><mo>↪</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'> i\;\colon\; E' \hookrightarrow E </annotation></semantics></math></div> <p>such that for each point <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> this is a linear embedding of fibers</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><msub><mo stretchy='false'>|</mo> <mi>x</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><msub><mo>′</mo> <mi>x</mi></msub><mo>↪</mo><msub><mi>E</mi> <mi>x</mi></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> i|_x \;\colon\; E'_x \hookrightarrow E_x \,. </annotation></semantics></math></div> <p>(This is a <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphism</a> in the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Vect</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Vect(X)</annotation></semantics></math> of topological vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.)</p> </div> <h3 id='TransitionFunctionsAndCechCohomology'>Transition functions and Cech cohomology</h3> <p>We discuss how topological vector bundles are equivalently given by <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycles</a> in <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech cohomology</a> constituted by their <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>transition functions</a>.</p> <div class='num_defn' id='ContinuousFunctionWithValuesInGLn'> <h6 id='definition_4'>Definition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> on <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> with values in the <a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a>)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>, regard the <a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>GL(n,k)</annotation></semantics></math> as a <a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a> with its standard <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topology</a>, given as the <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> via <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>⊂</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>≃</mo><msup><mi>k</mi> <mrow><mo stretchy='false'>(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy='false'>)</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>GL(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}</annotation></semantics></math> or as the or as the subspace topology <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>Maps</mi><mo stretchy='false'>(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>GL(n,k) \subset Maps(k^n, k^n)</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a> on the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a>. (That these topologies coincide is the statement of <a href='general+linear+group#AsSubspaceOfTheMappingSpace'>this prop.</a>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, we write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>U</mi><mo>↦</mo><msub><mi>Hom</mi> <mi>Top</mi></msub><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \underline{GL(n,k)} \;\colon\; U \mapsto Hom_{Top}(U, GL(n,k) ) </annotation></semantics></math></div> <p>for the assignment that sends an <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math> to the <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> of <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>U</mi><mo>→</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g \colon U \to GL(n,k)</annotation></semantics></math> (for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math> equipped with its <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>), regarded as a <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> via the pointwise group operation in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>GL(n,k)</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>g</mi> <mn>2</mn></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>x</mi><mo>↦</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>⋅</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> g_1 \cdot g_2 \;\colon\; x \mapsto g_1(x) \cdot g_2(x) \,. </annotation></semantics></math></div> <p>Moreover, for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>′</mo><mo>⊂</mo><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U' \subset U \subset X</annotation></semantics></math> an inclusion of open subsets, and for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>∈</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g \in \underline{GL(n,k)}(U)</annotation></semantics></math>, we write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><msub><mo stretchy='false'>|</mo> <mrow><mi>U</mi><mo>′</mo></mrow></msub><mo>∈</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>U</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> g|_{U'} \in \underline{GL(n,k)}(U') </annotation></semantics></math></div> <p>for the restriction of the continuous function from <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>U'</annotation></semantics></math>.</p> </div> <div class='num_remark'> <h6 id='remark_4'>Remark</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a> of <a class='existingWikiWord' href='/nlab/show/diff/group'>groups</a>)</strong></p> <p>In the language of <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a> the assignment <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder></mrow><annotation encoding='application/x-tex'>\underline{GL(n,k)}</annotation></semantics></math> from def. <a class='maruku-ref' href='#ContinuousFunctionWithValuesInGLn'>3</a> of continuous functions to open subsets and the restriction operations between these is called a <em><a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a> of groups on the <a class='existingWikiWord' href='/nlab/show/diff/category+of+open+subsets'>site of open subsets</a></em> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> </div> <div class='num_defn' id='TransitionFunctions'> <h6 id='definition_5'>Definition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>transition functions</a>)</strong></p> <p>Given a topological vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> as in def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a> and a choice of <a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>local trivialization</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>→</mo><mo>≃</mo></mover><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{\phi_i \colon U_i \times k^n \overset{\simeq}{\to} E|_{U_i}\}</annotation></semantics></math> (example <a class='maruku-ref' href='#TrivialTopologicalVectorBundle'>1</a>) there are for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i,j \in I</annotation></semantics></math> induced <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>⟶</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>}</mo></mrow> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \left\{ g_{i j} \;\colon\; (U_i \cap U_j) \longrightarrow GL(n, k) \right\}_{i,j \in I} </annotation></semantics></math></div> <p>to the <a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a> (as in def. <a class='maruku-ref' href='#ContinuousFunctionWithValuesInGLn'>3</a>) given by composing the local trivialization isomorphisms:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub></mrow></mover></mtd> <mtd><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>ϕ</mi> <mi>j</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub></mrow></mover></mtd> <mtd><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd /> <mtd><mrow><mo>(</mo><mi>x</mi><mo>,</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ (U_i \cap U_j) \times k^n &\overset{ \phi_i|_{U_i \cap U_j} }{\longrightarrow}& E|_{U_i \cap U_j} &\overset{ \phi_j^{-1}\vert_{U_i \cap U_j} }{\longrightarrow}& (U_i \cap U_j) \times k^n \\ (x,v) && \overset{\phantom{AAA}}{\mapsto} && \left( x, g_{i j}(x)(v) \right) } \,. </annotation></semantics></math></div> <p>These are called the <em><a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>transition functions</a></em> for the given local trivialization.</p> </div> <p>These functions satisfy a special property:</p> <div class='num_defn' id='CocycleCech'> <h6 id='definition_6'>Definition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech</a> <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycles</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>.</p> <p>A <em>normalized <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech cocycle</a> of degree 1 with <a class='existingWikiWord' href='/nlab/show/diff/coefficient'>coefficients</a></em> in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder></mrow><annotation encoding='application/x-tex'>\underline{GL(n,k)}</annotation></semantics></math> (def. <a class='maruku-ref' href='#ContinuousFunctionWithValuesInGLn'>3</a>) is</p> <ol> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math></p> </li> <li> <p>for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i,j \in I</annotation></semantics></math> a continuous function <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>→</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g_{i j} \colon U_i \cap U_j \to GL(n,k)</annotation></semantics></math> as in def. <a class='maruku-ref' href='#ContinuousFunctionWithValuesInGLn'>3</a></p> </li> </ol> <p>such that</p> <ol> <li> <p>(normalization) <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∀</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>=</mo><msub><mi>const</mi> <mn>1</mn></msub><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\forall}\left( g_{i i} = const_1 \right) </annotation></semantics></math> (the <a class='existingWikiWord' href='/nlab/show/diff/constant+function'>constant function</a> on the <a class='existingWikiWord' href='/nlab/show/diff/identity+element'>neutral element</a> in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>GL(n,k)</annotation></semantics></math>),</p> </li> <li> <p>(cocycle condition) <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∀</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></munder><mrow><mo>(</mo><msub><mi>g</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>⋅</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mtext>on</mtext><mspace width='thinmathspace' /><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>k</mi></msub><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>\underset{i,j \in I}{\forall}\left( g_{j k} \cdot g_{i j} = g_{i k}\;\;\text{on}\, U_i \cap U_j \cap U_k\right)</annotation></semantics></math>.</p> </li> </ol> <p>Write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> C^1(X, \underline{GL(n,k)} ) </annotation></semantics></math></div> <p>for the set of all such cocycles for given <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> and write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mi>GL</mi><mo>̲</mo></munder><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><munder><mo>⊔</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> C^1( X, \underline{GL}(k) ) \;\coloneqq\; \underset{n \in \mathbb{N}}{\sqcup} C^1(X, \underline{GL(n,k)}) </annotation></semantics></math></div> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> of all these cocycles as <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> varies.</p> </div> <div class='num_example' id='CocycleCechTransitionFunction'> <h6 id='example_4'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>transition functions</a> are <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech</a> <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycles</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> be a topological vector bundle (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>) and let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>→</mo><mo>≃</mo></mover><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{\phi_i \colon U_i \times k^n \overset{\simeq}{\to} E|_{U_{i}}\}_{i \in I}</annotation></semantics></math> be a local trivialization (example <a class='maruku-ref' href='#TrivialTopologicalVectorBundle'>1</a>).</p> <p>Then the set of induced <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>transition functions</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>→</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{g_{i j} \colon U_i \cap U_j \to GL(n)\}</annotation></semantics></math> according to def. <a class='maruku-ref' href='#TransitionFunctions'>4</a> is a <em>normalized Cech cocycle on <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with coefficients in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder></mrow><annotation encoding='application/x-tex'>\underline{GL(k)}</annotation></semantics></math></em>, according to def. <a class='maruku-ref' href='#CocycleCech'>5</a>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>This is immediate from the definition:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msub><mi>g</mi> <mrow><mi>i</mi><mi>i</mi></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><msub><mi>id</mi> <mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} g_{i i }(x) & = \phi_i^{-1} \circ \phi_i(x,-) \\ & = id_{k^n} \end{aligned} </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><msub><mi>g</mi> <mrow><mi>j</mi><mi>k</mi></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>⋅</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><msubsup><mi>ϕ</mi> <mi>k</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>ϕ</mi> <mi>j</mi></msub><mo>)</mo></mrow><mo>∘</mo><mrow><mo>(</mo><msubsup><mi>ϕ</mi> <mi>j</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><msubsup><mi>ϕ</mi> <mi>k</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>k</mi></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned} g_{j k}(x) \cdot g_{i j}(x) & = \left(\phi_k^{-1} \circ \phi_j\right) \circ \left(\phi_j^{-1}\circ \phi_i\right)(x,-) \\ & = \phi_k^{-1} \circ \phi_i(x,-) \\ & = g_{i k}(x) \end{aligned} \,. </annotation></semantics></math></div></div> <p>Conversely:</p> <div class='num_example' id='TopologicalVectorBundleFromCechCocycle'> <h6 id='example_5'>Example</h6> <p><strong>(topological vector bundle constructed from a <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech</a> <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> and let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c \in C^1(X, \underline{GL(k)})</annotation></semantics></math> a Cech cocycle on <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> according to def. <a class='maruku-ref' href='#CocycleCech'>5</a>, with open cover <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> and component functions <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{g_{i j}\}_{i,j \in I}</annotation></semantics></math>.</p> <p>This induces an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> on the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'> \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n </annotation></semantics></math></div> <p>(of the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> of the patches <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U_i \subset X</annotation></semantics></math> regarded as <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>topological subspaces</a> with the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mo stretchy='false'>{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy='false'>}</mo></mrow></munder><mi>k</mi></mrow><annotation encoding='application/x-tex'>k^n = \underset{\{1,\cdots, n\}}{\prod} k</annotation></semantics></math>) given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>∼</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>y</mi><mo>,</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>w</mi><mo stretchy='false'>)</mo><mo maxsize='1.2em' minsize='1.2em'>)</mo><mspace width='thickmathspace' /><mo>⇔</mo><mspace width='thickmathspace' /><mrow><mo>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mtext>and</mtext><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>w</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \big( ((x,i), v) \;\sim\; ((y,j), w) \big) \;\Leftrightarrow\; \left( (x = y) \;\text{and}\; (g_{i j}(x)(v) = w) \right) \,. </annotation></semantics></math></div> <p>Write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mrow><mo>(</mo><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>)</mo></mrow><mo stretchy='false'>/</mo><mrow><mo>(</mo><msub><mrow><mo>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> E(c) \;\coloneqq\; \left( \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \right) / \left( \left\{ g_{i j} \right\}_{i,j \in I} \right) </annotation></semantics></math></div> <p>for the resulting <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a>. This comes with the evident projection</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>π</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>[</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo>,</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>v</mi><mo stretchy='false'>]</mo></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mi>x</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ E(c) &\overset{\phantom{AA}\pi \phantom{AA}}{\longrightarrow}& X \\ [(x,i,),v] &\overset{\phantom{AAA}}{\mapsto}& x } </annotation></semantics></math></div> <p>which is a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> (by the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> construction, since the corresponding continuous <a class='existingWikiWord' href='/nlab/show/diff/function'>function</a> on the un-quotientd disjoint union space respects the equivalence relation). Moreover, each <a class='existingWikiWord' href='/nlab/show/diff/fiber'>fiber</a> of this map is identified with <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>k^n</annotation></semantics></math>, and hence canonicaly carries the structure of a <a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a>.</p> <p>Finally, the quotient co-projections constitute a local trivialization of this vector bundle over the given open cover.</p> <p>Therefore <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E(c) \to X</annotation></semantics></math> is a topological vector bundle (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>). We say it is the topological vector bundle <em>glued from the transition functions</em>.</p> </div> <div class='num_remark'> <h6 id='remark_5'>Remark</h6> <p><strong>(bundle glued from <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech</a> <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a> is a <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a>)</strong></p> <p>Stated more <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theoretically</a>, the constructure of a topological vector bundle from Cech cocycle data in example <a class='maruku-ref' href='#TopologicalVectorBundleFromCechCocycle'>5</a> is a <a href='Top#UniversalConstructions'>universal construction in topological spaces</a>, namely the <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a> of the two morphisms</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>,</mo><mi>μ</mi><mo>:</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mi>j</mi></mrow></munder><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>×</mo><mi>V</mi><mover><mo>→</mo><mo>→</mo></mover><munder><mo>⊔</mo><mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>i, \mu: \underset{i j}{\sqcup} (U_i \cap U_j) \times V \overset{\to}{\to} \underset{i}{\sqcup} U_i \times V</annotation></semantics></math></div> <p>in the category of vector space objects in the slice category <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi><mo stretchy='false'>/</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>Top/X</annotation></semantics></math>. Here the restriction of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> to the coproduct summands is induced by inclusion:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>×</mo><mi>V</mi><mo>↪</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><mi>V</mi><mo>↪</mo><munder><mo>⊔</mo><mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>(U_i \cap U_j) \times V \hookrightarrow U_i \times V \hookrightarrow \underset{i}{\sqcup} U_i \times V</annotation></semantics></math></div> <p>and the restriction of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>μ</mi></mrow><annotation encoding='application/x-tex'>\mu</annotation></semantics></math> to the coproduct summands is via the action of the transition functions:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>×</mo><mi>V</mi><mover><mo>→</mo><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>⟨</mo><mi>incl</mi><mo>,</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>⟩</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>V</mi></mrow></mover><msub><mi>U</mi> <mi>j</mi></msub><mo>×</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>V</mi><mover><mo>→</mo><mi>action</mi></mover><msub><mi>U</mi> <mi>j</mi></msub><mo>×</mo><mi>V</mi><mo>↪</mo><munder><mo>⊔</mo><mi>j</mi></munder><msub><mi>U</mi> <mi>j</mi></msub><mo>×</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>(U_i \cap U_j) \times V \overset{(\langle incl, g_{i j} \rangle) \times V}{\to} U_j \times GL(V) \times V \overset{action}{\to} U_j \times V \hookrightarrow \underset{j}{\sqcup} U_j \times V</annotation></semantics></math></div></div> <p>In fact, extracting transition functions from a vector bundle by def. <a class='maruku-ref' href='#TransitionFunctions'>4</a> and constructing a vector bundle from Cech coycle data as above are operations that are inverse to each other, up to <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>.</p> <div class='num_prop' id='FromTransitionFunctionsReconstructVectorBundle'> <h6 id='proposition'>Proposition</h6> <p><strong>(topological vector bundle reconstructed from its <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>transition functions</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>E</mi><mover><mo>→</mo><mi>π</mi></mover><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[E \overset{\pi}{\to} X]</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a> (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>), let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> of the base space, and let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></munderover><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>}</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\left\{ U_i \times k^n \underoverset{\simeq}{\phi_i}{\longrightarrow} E|_{U_i} \right\}_{i \in I}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>local trivialization</a>.</p> <p>Write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≔</mo><msubsup><mi>ϕ</mi> <mi>j</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>→</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>}</mo></mrow> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \left\{ g_{i j} \coloneqq \phi_j^{-1}\circ \phi_i \colon U_i \cap U_j \to GL(n,k) \right\}_{i,j \in I} </annotation></semantics></math></div> <p>for the corresponding <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>transition functions</a> (def. <a class='maruku-ref' href='#TransitionFunctions'>4</a>). Then there is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> of vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>)</mo></mrow><mo stretchy='false'>/</mo><mrow><mo>(</mo><msub><mrow><mo>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>)</mo></mrow><mspace width='thickmathspace' /><munderover><mo>⟶</mo><mo>≃</mo><mrow><mo stretchy='false'>(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></munderover><mspace width='thickmathspace' /><mi>E</mi></mrow><annotation encoding='application/x-tex'> \left( \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \right) / \left( \left\{ g_{i j} \right\}_{i,j \in I} \right) \;\underoverset{\simeq}{(\phi_i)_{i \in I}}{\longrightarrow}\; E </annotation></semantics></math></div> <p>from the vector bundle glued from the transition functions according to def. <a class='maruku-ref' href='#TransitionFunctions'>4</a> to the original bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>, whose components are the original local trivialization isomorphisms.</p> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> (<a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>), continuous functions out of them are equivalently sets of continuous functions out of every summand space. Hence the set of local trivializations <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><munderover><mo>→</mo><mo>≃</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></munderover><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>⊂</mo><mi>E</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \times k^n \underoverset{\simeq}{\phi_i}{\to} E|_{U_i} \subset E\}_{i \in I}</annotation></semantics></math> may be collected into a single <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mover><mi>E</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \underset{i \in I}{\sqcup} U_i \times k^n \overset{(\phi_i)_{i \in I}}{\longrightarrow } E \,. </annotation></semantics></math></div> <p>By construction this function respects the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> on the disjoint union space given by the transition functions, in that for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>x \in U_i \cap U_j</annotation></semantics></math> we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>ϕ</mi> <mi>j</mi></msub><mo>∘</mo><msubsup><mi>ϕ</mi> <mi>j</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>ϕ</mi> <mi>j</mi></msub><mo>∘</mo><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \phi_i((x,i),v) = \phi_j \circ \phi_j^{-1} \circ \phi_i((x,i),v) = \phi_j \circ ((x,j),g_{i j}(x)(v)) \,. </annotation></semantics></math></div> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a> coprojection this means that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>(\phi_i)_{i \in I}</annotation></semantics></math> uniquely <a class='existingWikiWord' href='/nlab/show/diff/extension'>extends</a> to a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> on the quotient space such that the following <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>diagram commutes</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo>∃</mo><mo>!</mo></mrow></msub></mtd></mtr> <mtr><mtd><mrow><mo>(</mo><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>)</mo></mrow><mo stretchy='false'>/</mo><mrow><mo>(</mo><msub><mrow><mo>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n &\overset{(\phi_i)_{i \in I}}{\longrightarrow}& E \\ \downarrow & \nearrow_{\exists !} \\ \left( \left( \underset{i \in I}{\sqcup} U_i \right) \times k^n \right) / \left( \left\{ g_{i j} \right\}_{i,j \in I} \right) } \,. </annotation></semantics></math></div> <p>It is clear that this continuous function is a <a class='existingWikiWord' href='/nlab/show/diff/bijection'>bijection</a>. Hence to show that it is a <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a>, it is now sufficient to show that this is an <a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a> (by <a href='Introduction+to+Topology+--+1#HomeoContinuousOpenBijection'>this prop.</a>).</p> <p>So let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>O</mi></mrow><annotation encoding='application/x-tex'>O</annotation></semantics></math> be an subset in the quotient space which is open. By definition of the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topology</a> this means equivalently that its restriction <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>O</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>O_i</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>U_i \times k^n</annotation></semantics></math> is open for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i \in I</annotation></semantics></math>. Since the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\phi_i</annotation></semantics></math> are homeomorphsms, it follows that the images <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><msub><mi>O</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>⊂</mo><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\phi_i(O_i) \subset E\vert_{U_ i}</annotation></semantics></math> are open. By the nature of the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, this means that these images are open also in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>. Therefore also the union <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>O</mi><mo stretchy='false'>)</mo><mo>=</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><msub><mi>O</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(O) = \underset{i \in I}{\cup} \phi_i(O_i)</annotation></semantics></math> is open.</p> </div> <div class='num_defn' id='CoboundaryCech'> <h6 id='definition_7'>Definition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/coboundary'>coboundary</a> between <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech</a> <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycles</a> )</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> and let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c_1, c_2 \in C^1(X, \underline{GL(k)})</annotation></semantics></math> be two <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech</a> <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycles</a> (def. <a class='maruku-ref' href='#CocycleCech'>5</a>), given by</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>U</mi><msub><mo>′</mo> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>′</mo><mo>∈</mo><mi>I</mi><mo>′</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U'_i \subset X\}_{i' \in I'}</annotation></semantics></math> two <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open covers</a>,</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>→</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo>,</mo><msub><mi>n</mi> <mo stretchy='false'>)</mo></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{g_{i j} \colon U_i \cap U_j \to GL(k,n_)\}_{i,j \in I}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>g</mi> <mo>′</mo></msub><msub><mo /><mrow><mi>i</mi><mo>′</mo><mo>,</mo><mi>j</mi><mo>′</mo></mrow></msub><mo lspace='verythinmathspace'>:</mo><mi>U</mi><msub><mo>′</mo> <mrow><mi>i</mi><mo>′</mo></mrow></msub><mo>∩</mo><mi>U</mi><msub><mo>′</mo> <mrow><mi>j</mi><mo>′</mo></mrow></msub><mo>→</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>′</mo><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>′</mo><mo>,</mo><mi>j</mi><mo>′</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>\{g_'_{i',j'} \colon U'_{i'} \cap U'_{j'} \to GL(n',k) \}_{i', j'}</annotation></semantics></math> the corrsponding component functions.</p> </li> </ol> <p>Then a <em><a class='existingWikiWord' href='/nlab/show/diff/coboundary'>coboundary</a></em> between these two cocycles is</p> <ol> <li> <p>the condition that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mi>n</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>n = n'</annotation></semantics></math>,</p> </li> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{V_\alpha \subset X\}_{\alpha \in A}</annotation></semantics></math>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/function'>functions</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>\phi \colon A \to I</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo>′</mo><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>J</mi></mrow><annotation encoding='application/x-tex'>\phi' \colon A \to J</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∀</mo><mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow></munder><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>⊂</mo><msub><mi>U</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><mo>)</mo></mrow><mspace width='thinmathspace' /><mtext>and</mtext><mspace width='thinmathspace' /><mrow><mo>(</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>⊂</mo><mi>U</mi><msub><mo>′</mo> <mrow><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>\underset{\alpha \in A}{\forall}\left( \left( V_\alpha \subset U_{\phi(\alpha)} \right) \,\text{and}\, \left( V_\alpha \subset U'_{\phi'(\alpha)} \right) \right)</annotation></semantics></math></p> </li> <li> <p>a set <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>κ</mi> <mi>α</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>→</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{ \kappa_\alpha \colon V_\alpha \to GL(n,k) \}</annotation></semantics></math> of continuous functions as in def. <a class='maruku-ref' href='#CocycleCech'>5</a></p> </li> </ol> <p>such that</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∀</mo><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi>A</mi></mrow></munder><mrow><mo>(</mo><msub><mi>κ</mi> <mi>β</mi></msub><mo>⋅</mo><msub><mi>g</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow></msub><mo>=</mo><mi>g</mi><msub><mo>′</mo> <mrow><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow></msub><mo>⋅</mo><msub><mi>κ</mi> <mi>α</mi></msub><mspace width='thinmathspace' /><mspace width='thinmathspace' /><mtext>on</mtext><mspace width='thinmathspace' /><mspace width='thinmathspace' /><msub><mi>V</mi> <mi>α</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>β</mi></msub><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>\underset{ \alpha, \beta \in A }{\forall} \left( \kappa_{\beta} \cdot g_{\phi(\alpha) \phi(\beta)} = g'_{\phi'(\alpha) \phi'(\beta)} \cdot \kappa_{\alpha} \,\, \text{on}\,\, V_\alpha \cap V_\beta \right) </annotation></semantics></math>,</p> <p>hence such that the following <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagrams</a> of <a class='existingWikiWord' href='/nlab/show/diff/linear+map'>linear maps</a> <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>commute</a> for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>\alpha, \beta \in A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>β</mi></msub></mrow><annotation encoding='application/x-tex'>x \in V_{\alpha} \cap V_\beta</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>g</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>κ</mi> <mi>α</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>κ</mi> <mi>β</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>g</mi><msub><mo>′</mo> <mrow><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></munder></mtd> <mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ k^n &\overset{ g_{\phi(\alpha) \phi(\beta)}(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{\kappa_{\alpha}(x)} }\downarrow && \downarrow^{\mathrlap{ \kappa_{\beta}(x) }} \\ k^n &\underset{ g'_{\phi'(\alpha) \phi'(\beta)}(x) }{\longrightarrow}& k^n } \,. </annotation></semantics></math></div></li> </ul> <p>Say that two Cech cocycles are <em>cohomologous</em> if there exists a coboundary between them.</p> </div> <div class='num_example' id='FinerCoverCech'> <h6 id='example_6'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/refinement'>refinement</a> of a <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech</a> <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a> is a <a class='existingWikiWord' href='/nlab/show/diff/coboundary'>coboundary</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> and let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c \in C^1(X, \underline{GL(k)})</annotation></semantics></math> be a Cech cocycle as in def. <a class='maruku-ref' href='#CocycleCech'>5</a>, with respect to some open cover <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> given by component functions <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{g_{i j}\}_{i,j \in I}</annotation></semantics></math>.</p> <p>Then for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{V_\alpha \subset X\}_{\alpha \in A}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/refinement'>refinement</a> of the given open cover, hence an open cover such that there exists a <a class='existingWikiWord' href='/nlab/show/diff/function'>function</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>\phi \colon A \to I</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∀</mo><mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow></munder><mrow><mo>(</mo><mi>V</mi><mi>α</mi><mo>⊂</mo><msub><mi>U</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>\underset{\alpha \in A}{\forall}\left( V\alpha \subset U_{\phi(\alpha)} \right)</annotation></semantics></math>, then</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_231' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><msub><mo>′</mo> <mrow><mi>α</mi><mi>β</mi></mrow></msub><mo>≔</mo><msub><mi>g</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>β</mi></msub><mo>⟶</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> g'_{ \alpha \beta } \coloneqq g_{\phi(\alpha) \phi(\beta)} \colon V_\alpha \cap V_\beta \longrightarrow GL(n,k) </annotation></semantics></math></div> <p>are the components of a Cech cocycle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>c'</annotation></semantics></math> which is cohomologous to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi></mrow><annotation encoding='application/x-tex'>c</annotation></semantics></math>.</p> </div> <div class='num_prop' id='CechCoboundaryFromIsomorphismBetweenVectoreBundles'> <h6 id='proposition_2'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> of topological vector bundles induces <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech</a> <a class='existingWikiWord' href='/nlab/show/diff/coboundary'>coboundary</a> between their <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>transition functions</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a topological space, and let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c_1, c_2 \in C^1(X, \underline{GL(n,k)} )</annotation></semantics></math> be two Cech cocycles as in def. <a class='maruku-ref' href='#CocycleCech'>5</a>.</p> <p>Every <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> of topological vector bundles</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> f \;\colon\; E(c_1) \overset{\simeq}{\longrightarrow} E(c_2) </annotation></semantics></math></div> <p>between the vector bundles glued from these cocycles according to def. <a class='maruku-ref' href='#TopologicalVectorBundleFromCechCocycle'>5</a> induces a coboundary between the two cocycles,</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>∼</mo><msub><mi>c</mi> <mn>2</mn></msub><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> c_1 \sim c_2 \,, </annotation></semantics></math></div> <p>according to def. <a class='maruku-ref' href='#CoboundaryCech'>6</a>.</p> </div> <div class='proof'> <h6 id='proof_4'>Proof</h6> <p>By example <a class='maruku-ref' href='#FinerCoverCech'>6</a> we may assume without restriction that the two Cech cocycles are defined with respect to the same open cover <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> (for if they are not, then both are cohomologous to cocycles on a joint refinement of the original covers and we may argue with these).</p> <p>Accordingly, by example <a class='maruku-ref' href='#TopologicalVectorBundleFromCechCocycle'>5</a> the two bundles <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E(c_1)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_240' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E(c_2)</annotation></semantics></math> both have local trivializations of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_241' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup></mrow></munderover><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> \{ U_i \times k^n \underoverset{\simeq}{\phi^1_i}{\longrightarrow} E(c_1)\vert_{U_i}\} </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_242' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>2</mn></msubsup></mrow></munderover><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> \{ U_i \times k^n \underoverset{\simeq}{\phi^2_i}{\longrightarrow} E(c_2)\vert_{U_i}\} </annotation></semantics></math></div> <p>over this cover. Consider then for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_243' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i \in I</annotation></semantics></math> the function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_244' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>≔</mo><mo stretchy='false'>(</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>2</mn></msubsup><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>f</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>∘</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> f_i \coloneqq (\phi_i^2)^{-1}\circ f\vert_{U_i} \circ \phi^1_i \,, </annotation></semantics></math></div> <p>hence the unique function making the following <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>diagram commute</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_245' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup></mrow></munderover></mtd> <mtd><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munderover><mo>⟶</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>2</mn></msubsup></mrow><mo>≃</mo></munderover></mtd> <mtd><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ U_i \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& E(c_1)\vert_{U_i} \\ {}^{\mathllap{f_i}}\downarrow && \downarrow^{\mathrlap{ f }} \\ U_i \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& E(c_2)\vert_{U_i} } \,. </annotation></semantics></math></div> <p>This induces for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_246' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i,j \in I</annotation></semantics></math> the following composite commuting diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_247' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup></mrow></munderover></mtd> <mtd><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><mo stretchy='false'>(</mo><msubsup><mi>ϕ</mi> <mi>j</mi> <mn>1</mn></msubsup><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow></munderover></mtd> <mtd><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mi>f</mi></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>f</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munderover><mo>⟶</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>2</mn></msubsup></mrow><mo>≃</mo></munderover></mtd> <mtd><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow></msub></mtd> <mtd><munderover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msubsup><mi>ϕ</mi> <mi>j</mi> <mn>2</mn></msubsup><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><mo>≃</mo></munderover></mtd> <mtd><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ (U_i \cap U_j) \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& E(c_1)\vert_{U_i \cap U_j} & \underoverset{\simeq}{(\phi^1_j)^{-1}}{\longrightarrow} & (U_i \cap U_j) \times k^n \\ {}^{\mathllap{f_i}}\downarrow && \downarrow^{\mathrlap{ f }} && \downarrow^{\mathrlap{ f_j }} \\ (U_i \cap U_j) \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& E(c_2)\vert_{U_1 \cap U_2} &\underoverset{(\phi^2_j)^{-1}}{\simeq}{\longrightarrow}& (U_i \cap U_j) \times k^n } \,. </annotation></semantics></math></div> <p>By construction, the two horizonal composites of this diagram are pointwise given by the components <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_248' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow> <mn>1</mn></msubsup></mrow><annotation encoding='application/x-tex'>g^1_{i j}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_249' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow> <mn>2</mn></msubsup></mrow><annotation encoding='application/x-tex'>g^2_{i j}</annotation></semantics></math>of the cocycles <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_250' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>c_1</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_251' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>c_2</annotation></semantics></math>, respectively. Hence the commutativity of this diagram is equivalently the commutativity of these diagrams:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_252' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow> <mn>1</mn></msubsup><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>f</mi> <mi>j</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munder><mo>⟶</mo><mrow><msubsup><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow> <mn>2</mn></msubsup><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></munder></mtd> <mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ k^n &\overset{ g^1_{i j}(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{ f_i(x) } }\downarrow && \downarrow^{\mathrlap{ f_j(x) }} \\ k^n &\underset{ g^2_{ i j }(x) }{\longrightarrow}& k^n } \,. </annotation></semantics></math></div> <p>for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_253' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i,j \in I</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_254' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>x \in U_i \cap U_j</annotation></semantics></math>. By def. <a class='maruku-ref' href='#CoboundaryCech'>6</a> this exhibits the required coboundary.</p> </div> <div class='num_defn' id='CohomologyCech'> <h6 id='definition_8'>Definition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech cohomology</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_255' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>. The relation <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_256' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>\sim</annotation></semantics></math> on <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech cocycles</a> of being cohomologous (def. <a class='maruku-ref' href='#CoboundaryCech'>6</a>) is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> on the set <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_257' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^1( X, \underline{GL(k)} )</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech</a> <a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycles</a> (def. <a class='maruku-ref' href='#CocycleCech'>5</a>).</p> <p>Write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_258' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'> H^1(X, \underline{GL(k)} ) \;\coloneqq\; C^1(X, \underline{GL(k)} )/\sim </annotation></semantics></math></div> <p>for the resulting set of <a class='existingWikiWord' href='/nlab/show/diff/equivalence+class'>equivalence classes</a>. This is called the <em><a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech cohomology</a> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_259' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> in degree 1 with <a class='existingWikiWord' href='/nlab/show/diff/coefficient'>coefficients</a> in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_260' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder></mrow><annotation encoding='application/x-tex'>\underline{GL(k)}</annotation></semantics></math>.</em></p> </div> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p><strong>(degree-1 <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech cohomology</a> computes <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundles</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_261' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>.</p> <p>The construction of gluing a topological vector bundle from a Cech cocycle (example <a class='maruku-ref' href='#TopologicalVectorBundleFromCechCocycle'>5</a>) constitutes a <a class='existingWikiWord' href='/nlab/show/diff/natural+bijection'>bijection</a> between the degree-1 Cech cohomology of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_262' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with coefficients in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_263' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>GL(n,k)</annotation></semantics></math> (def. <a class='maruku-ref' href='#CohomologyCech'>7</a>) and the set of <a class='existingWikiWord' href='/nlab/show/diff/isomorphism+class'>isomorphism classes</a> of topological vector bundles on <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_264' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>, remark <a class='maruku-ref' href='#TopologicalVectorBundlesCategory'>1</a>):</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_265' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mo>≃</mo><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>Vect</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>/</mo><mo>∼</mo></mrow></msub></mtd></mtr> <mtr><mtd><mi>c</mi></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mi>E</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ H^1(X,\underline{GL(k)}) &\overset{\phantom{AA}\simeq \phantom{AA}}{\longrightarrow}& Vect(X)_{/\sim} \\ c &\overset{\phantom{AAA}}{\mapsto}& E(c) } \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_5'>Proof</h6> <p>First we need to see that the function is well defined, hence that if cocycles <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_266' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><msup><mi>C</mi> <mn>1</mn></msup><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><munder><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><mo>̲</mo></munder><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>c_1, c_2 \in C^1(X,\underline{GL(k)})</annotation></semantics></math> are related by a coboundary, <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_267' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>∼</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>c_1 \sim c_2</annotation></semantics></math> (def. <a class='maruku-ref' href='#CoboundaryCech'>6</a>), then the vector bundles <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_268' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E(c_1)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_269' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E(c_2)</annotation></semantics></math> are related by an isomorphism.</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_270' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{V_\alpha \subset X\}_{\alpha \in A}</annotation></semantics></math> be the open cover with respect to which the coboundary <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_271' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>κ</mi> <mi>α</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>→</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><msub><mo stretchy='false'>}</mo> <mi>α</mi></msub></mrow><annotation encoding='application/x-tex'>\{\kappa_\alpha \colon V_\alpha \to GL(n,k)\}_{\alpha}</annotation></semantics></math> is defined, with refining functions <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_272' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>\phi \colon A \to I</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_273' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo>′</mo><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>I</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>\phi' \colon A \to I'</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_274' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><munderover><mo>→</mo><mo>≃</mo><mrow><msub><mi>ψ</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub></mrow></munderover><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub><mo>}</mo></mrow> <mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\left\{ V_\alpha \times k^n \underoverset{\simeq}{\psi_{\phi(\alpha)}\vert_{V_\alpha} }{\to} E(c_1)\vert_{V_\alpha} \right\}_{\alpha \in A}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_275' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><munderover><mo>→</mo><mo>≃</mo><mrow><mi>ψ</mi><msub><mo>′</mo> <mrow><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub></mrow></munderover><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub><mo>}</mo></mrow> <mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\left\{ V_\alpha \times k^n \underoverset{\simeq}{\psi'_{\phi'(\alpha)}\vert_{V_\alpha} }{\to} E(c_2)\vert_{V_\alpha} \right\}_{\alpha \in A}</annotation></semantics></math> be the corresponding restrictions of the canonical local trivilizations of the two glued bundles.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_276' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>\alpha \in A</annotation></semantics></math> define</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_277' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>α</mi></msub><mo>≔</mo><mi>ψ</mi><msub><mo>′</mo> <mrow><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub><mo>∘</mo><msub><mi>κ</mi> <mi>α</mi></msub><mo>∘</mo><mo stretchy='false'>(</mo><msub><mi>ψ</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mphantom><mi>AAAA</mi></mphantom><mtext>hence:</mtext><mphantom><mi>AAA</mi></mphantom><mrow><mtable><mtr><mtd><msub><mi>V</mi> <mi>α</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ψ</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub></mrow></mover></mtd> <mtd><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>κ</mi> <mi>α</mi></msub></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>f</mi> <mi>α</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>V</mi> <mi>α</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟵</mo><mrow><mo stretchy='false'>(</mo><mi>ψ</mi><msub><mo>′</mo> <mrow><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow></mover></mtd> <mtd><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f_\alpha \coloneqq \psi'_{\phi'(\alpha)}\vert_{V_\alpha} \circ \kappa_\alpha \circ (\psi_{\phi(\alpha)}\vert_{V_\alpha} )^{-1} \phantom{AAAA} \text{hence:} \phantom{AAA} \array{ V_\alpha \times k^n &\overset{ \psi_{\phi(\alpha)}\vert_{V_\alpha} }{\longrightarrow}& E(c_1)\vert_{V_\alpha} \\ {}^{\mathllap{\kappa_\alpha}}\downarrow && \downarrow^{\mathrlap{f_\alpha}} \\ V_\alpha \times k^n &\overset{ (\psi'_{\phi'(\alpha)}\vert_{V_\alpha})^{-1} }{\longleftarrow}& E(c_1)\vert_{V_\alpha} } \,. </annotation></semantics></math></div> <p>Observe that for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_278' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>\alpha, \beta \in A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_279' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>V</mi> <mi>α</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>β</mi></msub></mrow><annotation encoding='application/x-tex'>x \in V_\alpha \cap V_\beta</annotation></semantics></math> the coboundary condition implies that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_280' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>α</mi></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>β</mi></msub></mrow></msub><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><msub><mi>f</mi> <mi>β</mi></msub><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>β</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'> f_\alpha\vert_{V_\alpha \cap V_\beta} \;=\; f_\beta\vert_{V_\alpha \cap V_\beta} </annotation></semantics></math></div> <p>because in the diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_281' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>g</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>κ</mi> <mi>α</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>κ</mi> <mi>β</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>g</mi><msub><mo>′</mo> <mrow><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></munder></mtd> <mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mphantom><mi>AAAAA</mi></mphantom><mo>=</mo><mphantom><mi>AAAAA</mi></mphantom><mrow><mtable><mtr><mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>ψ</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><msub><mo stretchy='false'>)</mo> <mi>x</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msub><mi>ψ</mi> <mrow><mi>ϕ</mi><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow></msub><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>κ</mi> <mi>α</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><mo>∃</mo><mo>!</mo></mrow></mpadded></msup></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>β</mi> <mi>α</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>ψ</mi><msub><mo>′</mo> <mrow><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><msub><mo stretchy='false'>)</mo> <mi>x</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><mi>ψ</mi><msub><mo>′</mo> <mrow><mi>ϕ</mi><mo>′</mo><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow></msub><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></mover></mtd> <mtd><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ k^n &\overset{ g_{\phi(\alpha) \phi(\beta) }(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{\kappa_\alpha(x)}}\downarrow && \downarrow^{\mathrlap{\kappa_{\beta}(x)}} \\ k^n &\underset{g'_{\phi'(\alpha) \phi'(\beta)}(x) }{\longrightarrow}& k^n } \phantom{AAAAA} = \phantom{AAAAA} \array{ k^n &\overset{ \psi_{\phi(\alpha)}(x) }{\longrightarrow}& E(c_1)_x &\overset{ (\psi_{\phi(\beta)})^{-1}(x) }{\longrightarrow}& k^n \\ {}^{\mathllap{\kappa_\alpha(x)}}\downarrow && \downarrow^{\mathrlap{\exists !} } && \downarrow^{\mathrlap{\beta_\alpha(x)}} \\ k^n &\overset{ \psi'_{\phi'(\alpha)}(x) }{\longrightarrow}& E(c_2)_x &\overset{ (\psi'_{\phi'(\beta)})^{-1}(x) }{\longrightarrow}& k^n } </annotation></semantics></math></div> <p>the vertical morphism in the middle on the right is unique, by the fact that all other morphisms in the diagram on the right are invertible.</p> <p>Therefore there is a unique vector bundle homomorphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_282' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>→</mo><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> f\;\colon\; E(c_1) \to E(c_2) </annotation></semantics></math></div> <p>given for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_283' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>\alpha \in A</annotation></semantics></math> by <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_284' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub><mo>=</mo><msub><mi>f</mi> <mi>α</mi></msub></mrow><annotation encoding='application/x-tex'>f\vert_{V_\alpha} = f_\alpha</annotation></semantics></math>. Similarly there is a unique vector bundle homomorphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_285' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>→</mo><mi>E</mi><mo stretchy='false'>(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> f^{-1}\;\colon\; E(c_2) \to E(c_1) </annotation></semantics></math></div> <p>given for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_286' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>\alpha \in A</annotation></semantics></math> by <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_287' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>V</mi> <mi>α</mi></msub></mrow></msub><mo>=</mo><msubsup><mi>f</mi> <mi>α</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>f^{-1}\vert_{V_\alpha} = f^{-1}_\alpha</annotation></semantics></math>. Hence this is the required vector bundle isomorphism.</p> <p>Finally to see that the function from Cech cohomology classes to isomorphism classes of vector bundles thus defined is a bijection:</p> <p>By prop. <a class='maruku-ref' href='#FromTransitionFunctionsReconstructVectorBundle'>1</a> the function is <a class='existingWikiWord' href='/nlab/show/diff/surjection'>surjective</a>, and by prop. <a class='maruku-ref' href='#CechCoboundaryFromIsomorphismBetweenVectoreBundles'>2</a> it is injective.</p> </div> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_288' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></p> <h2 id='Examples'>Examples</h2> <div class='num_example' id='TautologicalLineBundle'> <h6 id='example_7'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/tautological+line+bundle'>tautological line bundle</a>)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_289' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> then the <a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_290' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><msup><mi>P</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>k P^n</annotation></semantics></math> carries the <em><a class='existingWikiWord' href='/nlab/show/diff/tautological+line+bundle'>tautological line bundle</a></em> whose fiber over the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_291' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-line <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_292' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>v</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>k</mi><msup><mi>P</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>[v] \in k P^n</annotation></semantics></math> is that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_293' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-line.</p> <p>For details see <a href='tautological+line+bundle#AsAtopologicalLieBundle'>there</a></p> </div> <div class='num_example'> <h6 id='example_8'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>)</strong></p> <p>Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_294' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>=</mo><mrow><mo>{</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo stretchy='false'>|</mo><mspace width='thickmathspace' /><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>}</mo></mrow><mspace width='thickmathspace' /><mo>⊂</mo><mspace width='thinmathspace' /><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'> S^1 = \left\{ (x,y) \;\vert\; x^2 + y^2 = 1 \right\} \;\subset\, \mathbb{R}^2 </annotation></semantics></math></div><div style='float: right; margin: 0 10px 10px 0;'> <img src='https://ncatlab.org/nlab/files/cylinder.jpg' width='190' /> </div> <p>be the <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a> with its <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>.</p> <p>Then the <a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial</a> <a class='existingWikiWord' href='/nlab/show/diff/real+vector+bundle'>real</a> <a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>line bundle</a> on the circle is the the <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_295' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'> S^1 \times \mathbb{R} </annotation></semantics></math></div></div> <div class='num_example' id='MoebiusStrip'> <h6 id='example_9'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a>)</strong></p> <p>Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_296' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>=</mo><mrow><mo>{</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo stretchy='false'>|</mo><mspace width='thickmathspace' /><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>}</mo></mrow><mspace width='thickmathspace' /><mo>⊂</mo><mspace width='thinmathspace' /><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'> S^1 = \left\{ (x,y) \;\vert\; x^2 + y^2 = 1 \right\} \;\subset\, \mathbb{R}^2 </annotation></semantics></math></div> <p>be the <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a> with its <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>. Consider the <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_297' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mi>n</mi></msub><mo>⊂</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>}</mo></mrow> <mrow><mi>n</mi><mo>∈</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>}</mo></mrow></msub></mrow><annotation encoding='application/x-tex'> \left\{ U_n \subset S^1 \right\}_{n \in \{0,1,2\}} </annotation></semantics></math></div> <p>with</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_298' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>n</mi></msub><mo>≔</mo><mrow><mo>{</mo><mo stretchy='false'>(</mo><mi>cos</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>sin</mi><mo stretchy='false'>(</mo><mi>β</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo stretchy='false'>|</mo><mspace width='thickmathspace' /><mi>n</mi><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo>−</mo><mi>ϵ</mi><mo><</mo><mi>α</mi><mo><</mo><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo>+</mo><mi>ϵ</mi><mo>}</mo></mrow></mrow><annotation encoding='application/x-tex'> U_n \coloneqq \left\{ (cos(\alpha), sin(\beta)) \;\vert\; n \frac{2 \pi }{3} - \epsilon \lt \alpha \lt (n+1) \frac{2\pi }{3} + \epsilon \right\} </annotation></semantics></math></div> <p>for any <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_299' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>π</mi><mo stretchy='false'>/</mo><mn>6</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\epsilon \in (0,2\pi/6)</annotation></semantics></math>.</p> <p>Define a <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech cohomology</a> cocycle (remark \ref{CechCoycleCondition}) on this cover by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_300' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><msub><mi>n</mi> <mn>2</mn></msub></mrow></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>const</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo stretchy='false'>|</mo></mtd> <mtd><mo stretchy='false'>(</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msub><mi>const</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msub></mtd> <mtd><mo stretchy='false'>|</mo></mtd> <mtd><mo stretchy='false'>(</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msub><mi>const</mi> <mn>1</mn></msub></mtd> <mtd><mo stretchy='false'>|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding='application/x-tex'> g_{n_1 n_2} = \left\{ \array{ const_{-1} & \vert & (n_1,n_2) = (0,2) \\ const_{-1} &\vert& (n_1,n_2) = (2,0) \\ const_1 &\vert& \text{otherwise} } \right. </annotation></semantics></math></div><div style='float: right; margin: 0 10px 10px 0;'> <img src='https://ncatlab.org/nlab/files/moebiusstrip.jpg' width='200' /> </div> <p>Since there are no non-trivial triple intersections, all cocycle conditions are evidently satisfied.</p> <p>Accordingly by example <a class='maruku-ref' href='#TopologicalVectorBundleFromCechCocycle'>5</a> these functions define a vector bundle. This is the <em><a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a></em></p> </div> <div class='num_example'> <h6 id='example_10'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/basic+complex+line+bundle+on+the+2-sphere'>basic complex line bundle on the 2-sphere</a>)</strong></p> <p>Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_301' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>≔</mo><mrow><mo>{</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo stretchy='false'>|</mo><mspace width='thickmathspace' /><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>z</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>}</mo></mrow><mo>⊂</mo><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding='application/x-tex'> S^2 \coloneqq \left\{ (x,y,z) \;\vert\; x^2 + y^2 + z^2 = 1 \right\} \subset \mathbb{R}^3 </annotation></semantics></math></div> <p>be the <a class='existingWikiWord' href='/nlab/show/diff/2-sphere'>2-sphere</a> with its <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>. Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_302' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>}</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mo stretchy='false'>{</mo><mo lspace='verythinmathspace' rspace='0em'>+</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>}</mo></mrow></msub></mrow><annotation encoding='application/x-tex'> \left\{ U_{i} \subset S^2 \right\}_{i \in \{+,-\}} </annotation></semantics></math></div> <p>be the two <a class='existingWikiWord' href='/nlab/show/diff/complement'>complements</a> of antipodal points</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_303' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mo>±</mo></msub><mo>≔</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy='false'>{</mo><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>±</mo><mn>1</mn><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> U_\pm \coloneqq S^2 \setminus \{(0, 0, \pm 1)\} \,. </annotation></semantics></math></div> <p>Define continuous functions</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_304' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>U</mi> <mo>+</mo></msub><mo>∩</mo><msub><mi>U</mi> <mo>−</mo></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>g</mi> <mrow><mo>±</mo><mo>∓</mo></mrow></msub></mrow></mover></mtd> <mtd><mi>GL</mi><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mi>ℂ</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>z</mi> <mn>2</mn></msup></mrow></msqrt><mspace width='thinmathspace' /><mi>cos</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mo>,</mo><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>z</mi> <mn>2</mn></msup></mrow></msqrt><mspace width='thinmathspace' /><mi>sin</mi><mo stretchy='false'>(</mo><mi>α</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>exp</mi><mo stretchy='false'>(</mo><mo>±</mo><mn>2</mn><mi>π</mi><mi>i</mi><mi>α</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ U_+ \cap U_- &\overset{g_{\pm \mp}}{\longrightarrow}& GL(1,\mathbb{C}) \\ ( \sqrt{1-z^2} \, cos(\alpha), \sqrt{1-z^2} \, sin(\alpha), z) &\mapsto& \exp(\pm 2\pi i \alpha) } \,. </annotation></semantics></math></div> <p>Since there are no non-trivial triple intersections, the only cocycle condition is</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_305' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mrow><mo>∓</mo><mo>±</mo></mrow></msub><msub><mi>g</mi> <mrow><mo>±</mo><mo>∓</mo></mrow></msub><mo>=</mo><msub><mi>g</mi> <mrow><mo>±</mo><mo>±</mo></mrow></msub><mo>=</mo><mi>id</mi></mrow><annotation encoding='application/x-tex'> g_{\mp \pm} g_{\pm \mp} = g_{\pm \pm} = id </annotation></semantics></math></div> <p>which is clearly satisfied.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/line+bundle'>complex line bundle</a> this defined is called the <em><a class='existingWikiWord' href='/nlab/show/diff/basic+complex+line+bundle+on+the+2-sphere'>basic complex line bundle on the 2-sphere</a></em>.</p> <p>With the 2-sphere identified with the <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex projective space</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_306' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{C} P^1</annotation></semantics></math> (the <a class='existingWikiWord' href='/nlab/show/diff/Riemann+sphere'>Riemann sphere</a>), the basic complex line bundle is the <a class='existingWikiWord' href='/nlab/show/diff/tautological+line+bundle'>tautological line bundle</a> (example <a class='maruku-ref' href='#TautologicalLineBundle'>7</a>) on <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_307' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{C}P^1</annotation></semantics></math>.</p> </div> <div class='num_example' id='ClutchingConstruction'> <h6 id='example_11'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/clutching+construction'>clutching construction</a>)</strong></p> <p>Generally, for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_308' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_309' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n \geq 1</annotation></semantics></math> then the <a class='existingWikiWord' href='/nlab/show/diff/sphere'>n-sphere</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_310' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math> may be covered by two open <a class='existingWikiWord' href='/nlab/show/diff/hemisphere'>hemispheres</a> intersecting in an <a class='existingWikiWord' href='/nlab/show/diff/equator'>equator</a> of the form <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_311' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>ϵ</mi><mo>,</mo><mi>ϵ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>S^{n-1} \times (-\epsilon, \epsilon)</annotation></semantics></math>. A vector bundle is then defined by specifying a single function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_312' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>+</mo><mo>−</mo></mrow></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⟶</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> g_{+-} \;\colon\; S^{n-1} \longrightarrow GL(n,k) \,. </annotation></semantics></math></div> <p>This is called the <em><a class='existingWikiWord' href='/nlab/show/diff/clutching+construction'>clutching construction</a></em> of vector bundles over <a class='existingWikiWord' href='/nlab/show/diff/sphere'>n-spheres</a>.</p> </div> <div class='num_example'> <h6 id='example_12'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a>)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_313' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> underlyithe ng a <a class='existingWikiWord' href='/nlab/show/diff/differentiable+manifold'>differentiable manifold</a> then its <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_314' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>T X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/real+vector+bundle'>real vector bundle</a> over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_315' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> whose <a class='existingWikiWord' href='/nlab/show/diff/rank'>rank</a> is the <a class='existingWikiWord' href='/nlab/show/diff/dimension'>dimension</a> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_316' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> </div> <div class='num_example'> <h6 id='example_13'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/normal+bundle'>normal bundle</a>)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_317' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mi>X</mi><mo>↪</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>i X \hookrightarrow Y</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+differentiable+manifolds'>embedding of differentiable manifolds</a>, then the <em><a class='existingWikiWord' href='/nlab/show/diff/normal+bundle'>normal bundle</a></em></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_318' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>i</mi></msub><mi>X</mi><mo>≔</mo><mi>T</mi><mi>Y</mi><mo stretchy='false'>/</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'> N_i X \coloneqq T Y/T X </annotation></semantics></math></div> <p>is the real vector bundle over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_319' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> whose <a class='existingWikiWord' href='/nlab/show/diff/fiber'>fiber</a> at <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_320' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/quotient+module'>quotient vector space</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_321' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>N</mi> <mi>i</mi></msub><mi>X</mi><msub><mo stretchy='false'>)</mo> <mi>x</mi></msub><mo>≔</mo><msub><mi>T</mi> <mrow><mi>i</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><mi>Y</mi><mo stretchy='false'>/</mo><msub><mi>T</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>(N_i X)_x \coloneqq T_{i(x)} Y / T_x X</annotation></semantics></math>.</p> </div> <h2 id='properties'>Properties</h2> <h3 id='BasicProperties'>Basic properties</h3> <div class='num_lemma' id='FiberwiseIsoisIsomorphismOfVectorBundles'> <h6 id='lemma'>Lemma</h6> <p><strong>(homomorphism of vector bundles is isomorphism as soon as it is a fiberwise isomorphism)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_322' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>E</mi> <mn>1</mn></msub><mo>→</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[E_1 \to X]</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_323' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo>→</mo><mi>X</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[E_2 \to X]</annotation></semantics></math> be two topological vector bundles (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>).</p> <p>If a <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> of vector bundles <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_324' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><msub><mi>E</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>f \colon E_1 \longrightarrow E_2</annotation></semantics></math> restricts on the <a class='existingWikiWord' href='/nlab/show/diff/fiber'>fiber</a> over each point to a linear isomorphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_325' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><msub><mo stretchy='false'>|</mo> <mi>x</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>)</mo> <mi>x</mi></msub><mover><mo>⟶</mo><mo>≃</mo></mover><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>)</mo> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'> f\vert_x \;\colon\; (E_1)_x \overset{\simeq}{\longrightarrow} (E_2)_x </annotation></semantics></math></div> <p>then <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_326' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is already an isomorphism of vector bundles.</p> </div> <div class='proof'> <h6 id='proof_6'>Proof</h6> <p>It is clear that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_327' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> has an <a class='existingWikiWord' href='/nlab/show/diff/inverse+function'>inverse function</a> of underlying sets <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_328' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo lspace='verythinmathspace'>:</mo><msub><mi>E</mi> <mn>2</mn></msub><msub><mo>→</mo> <mi>E</mi></msub><msub><mo /><mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>f^{-1} \colon E_2 \to _E_1</annotation></semantics></math> which is a function over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_329' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>: Over each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_330' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> it it the linear inverse <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_331' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><msub><mo stretchy='false'>|</mo> <mi>x</mi></msub><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>)</mo> <mi>x</mi></msub><mo>→</mo><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>)</mo> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>(f\vert_x)^{-1} \colon (E_2)_x \to (E_1)_x</annotation></semantics></math>.</p> <p>What we need to show is that this is a continuous function.</p> <p>By remark <a class='maruku-ref' href='#CommonOpenCoverLocalTrivialization'>3</a> we find an open cover <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_332' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> over which both bundles have a local trivialization.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_333' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mi>i</mi></msub><munderover><mo>→</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup></mrow></munderover><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>}</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mphantom><mi>AA</mi></mphantom><mtext>and</mtext><mphantom><mi>AA</mi></mphantom><msub><mrow><mo>{</mo><msub><mi>U</mi> <mi>i</mi></msub><munderover><mo>→</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>2</mn></msubsup></mrow></munderover><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>}</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \left\{ U_i \underoverset{\simeq}{\phi^1_i}{\to} (E_1)\vert_{U_i}\right\}_{i \in I} \phantom{AA} \text{and} \phantom{AA} \left\{ U_i \underoverset{\simeq}{\phi^2_i}{\to} (E_2)\vert_{U_i} \right\}_{i \in I} \,. </annotation></semantics></math></div> <p>Restricted to any patch <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_334' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i \in I</annotation></semantics></math> of this cover, the homomorphism <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_335' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>f|_{U_i}</annotation></semantics></math> induces a homomorphism of <a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial vector bundles</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_336' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>≔</mo><msubsup><mi>ϕ</mi> <mi>j</mi> <mn>2</mn></msubsup><msup><mo /><mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>f</mi><mo>∘</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup><mphantom><mi>AAAAAA</mi></mphantom><mrow><mtable><mtr><mtd><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup></mrow></munderover></mtd> <mtd><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd><msup><mrow /> <mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><mi>f</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munderover><mo>⟶</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>2</mn></msubsup></mrow><mo>≃</mo></munderover></mtd> <mtd><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f_i \coloneqq \phi^2_j^{-1} \circ f \circ \phi^1_i \phantom{AAAAAA} \array{ U_i \times k^n &\underoverset{\simeq}{\phi^1_i}{\longrightarrow}& (E_1)\vert|_{U_i} \\ {}^{f_i}\downarrow && \downarrow^{\mathrlap{f\vert_{U_i}}} \\ U_i \times k^n &\underoverset{\phi^2_i}{\simeq}{\longrightarrow}& (E_2)\vert_{U_j} } \,. </annotation></semantics></math></div> <p>Also the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_337' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>f_i</annotation></semantics></math> are fiberwise invertible, hence are continuous bijections. We claim that these are <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphisms</a>, hence that their inverse functions <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_338' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>(f_i)^{-1}</annotation></semantics></math> are also continuous.</p> <p>To this end we re-write the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_339' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>f_i</annotation></semantics></math> a little. First observe that by the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> and since they fix the base space <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_340' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math>, the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_341' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>f_i</annotation></semantics></math> are equivalently given by a continuous function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_342' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>⟶</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'> h_i \;\colon\; U_i \times k^n \longrightarrow k^n </annotation></semantics></math></div> <p>as</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_343' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><msub><mi>h</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f_i(x,v) = (x, h_i(x,v)) \,. </annotation></semantics></math></div> <p>Moreovern since <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_344' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>k^n</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a> (like every <a class='existingWikiWord' href='/nlab/show/diff/finite-dimensional+vector+space'>finite dimensional vector space</a>, by the <a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a>), the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a> <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> says (by <a href='Introduction+to+Topology+--+1#UniversalPropertyOfMappingSpace'>this prop.</a>) that there is a continuous function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_345' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>h</mi><mo stretchy='false'>˜</mo></mover> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>U</mi> <mi>i</mi></msub><mo>⟶</mo><mi>Maps</mi><mo stretchy='false'>(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \tilde h_i \;\colon\; U_i \longrightarrow Maps(k^n, k^n) </annotation></semantics></math></div> <p>(with <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_346' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Maps(k^n,k^n)</annotation></semantics></math> the set of continuous functions <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_347' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>k^n \to k^n</annotation></semantics></math> equipped with the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>) which factors <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_348' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>h_i</annotation></semantics></math> via the <a class='existingWikiWord' href='/nlab/show/diff/evaluation+map'>evaluation</a> map as</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_349' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mrow><msub><mover><mi>h</mi><mo stretchy='false'>˜</mo></mover> <mi>i</mi></msub><mo>×</mo><msub><mi>id</mi> <mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow></msub></mrow></mover><mi>Maps</mi><mo stretchy='false'>(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mi>ev</mi></mover><msup><mi>k</mi> <mi>n</mi></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> h_i \;\colon\; U_i \times k^n \overset{\tilde h_i \times id_{k^n}}{\longrightarrow} Maps(k^n, k^n) \times k^n \overset{ev}{\longrightarrow} k^n \,. </annotation></semantics></math></div> <p>By assumption of fiberwise linearity the functions <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_350' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>h</mi><mo stretchy='false'>˜</mo></mover> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\tilde h_i</annotation></semantics></math> in fact take values in the <a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_351' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>Maps</mi><mo stretchy='false'>(</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> GL(n,k) \subset Maps(k^n, k^n) </annotation></semantics></math></div> <p>and this inclusion is a <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a> onto its image (by <a href='general+linear+group#AsSubspaceOfTheMappingSpace'>this prop.</a>).</p> <p>Since passing to <a class='existingWikiWord' href='/nlab/show/diff/inverse+matrix'>inverse matrices</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_352' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>⟶</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (-)^{-1} \;\colon\; GL(n,k) \longrightarrow GL(n,k) </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/rational+function'>rational function</a> on its domain <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_353' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>⊂</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>≃</mo><msup><mi>k</mi> <mrow><mo stretchy='false'>(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy='false'>)</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>GL(n,k) \subset Mat_{n \times n}(k) \simeq k^{(n^2)}</annotation></semantics></math> inside <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> and since <a class='existingWikiWord' href='/nlab/show/diff/real+rational+functions+are+pointwise+continuous'>rational functions are continuous</a> on their domain of definition, it follows that the inverse of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_354' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>f_i</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_355' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><mi>id</mi><mo>,</mo><msub><mover><mi>h</mi><mo stretchy='false'>˜</mo></mover> <mi>i</mi></msub><mo stretchy='false'>)</mo></mrow></mover><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>×</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mrow><mi>id</mi><mo>×</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow></mover><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>×</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mrow><mi>id</mi><mo>×</mo><mi>ev</mi></mrow></mover><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'> (f_i)^{-1} \;\colon\; U_i \times k^n \overset{(id , \tilde h_i ) }{\longrightarrow} U_i \times k^n \times GL(n,k) \overset{ id \times (-)^{-1} }{\longrightarrow} U_i \times k^n \times GL(n,k) \overset{id \times ev}{\longrightarrow} U_i \times k^n </annotation></semantics></math></div> <p>is a continuous function.</p> <p>To conclude that also <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_356' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>f^{-1}</annotation></semantics></math> is a continuous function we make use prop. <a class='maruku-ref' href='#FromTransitionFunctionsReconstructVectorBundle'>1</a> to find an isomorphism between <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_357' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>E_2</annotation></semantics></math> and a <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_358' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub><mo>≃</mo><mrow><mo>(</mo><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><mo>)</mo></mrow><mo stretchy='false'>/</mo><mrow><mo>(</mo><msub><mrow><mo>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>}</mo></mrow> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> E_2 \simeq \left(\underset{i \in I}{\sqcup} (U_i \times k^n) \right) / \left( \left\{ g_{i j}\right\}_{i,j\in I} \right) \,. </annotation></semantics></math></div> <p>Hence <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_359' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>f^{-1}</annotation></semantics></math> is equivalently a function on this quotient space, and we need to show that as such it is continuous.</p> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> (the <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>) the set of continuous functions</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_360' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msubsup><mi>f</mi> <mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup></mrow></mover><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup></mrow></mover><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \{ U_i \times k^n \overset{f_i^{-1}}{\to} U_i \times k^n \overset{\phi^1_i}{\to} E_1 \}_{i \in I} </annotation></semantics></math></div> <p>corresponds to a single continuous function of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_361' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup><mo>∘</mo><msubsup><mi>f</mi> <mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>⟶</mo><msub><mi>E</mi> <mn>1</mn></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (\phi^1_i \circ f_i^{-1})_{i \in I} \;\colon\; \underset{i \in I}{\sqcup} U_i \times k^n \longrightarrow E_1 \,. </annotation></semantics></math></div> <p>These functions respect the equivalence relation, since for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_362' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>x \in U_i \cap U_j</annotation></semantics></math> we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_363' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup><mo>∘</mo><msubsup><mi>f</mi> <mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><msubsup><mi>ϕ</mi> <mi>j</mi> <mn>1</mn></msubsup><mo>∘</mo><msubsup><mi>f</mi> <mi>j</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>j</mi><mo stretchy='false'>)</mo><mo>,</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mphantom><mi>AAAA</mi></mphantom><mtext>since:</mtext><mphantom><mi>AAAA</mi></mphantom><mrow><mtable><mtr><mtd /> <mtd /> <mtd><msub><mi>E</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup><mo>∘</mo><msubsup><mi>f</mi> <mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow></mpadded></msup></mtd> <mtd><msup><mo>↖</mo> <mpadded width='0'><mrow><msubsup><mi>ϕ</mi> <mi>j</mi> <mn>1</mn></msubsup><mo>∘</mo><msubsup><mi>f</mi> <mi>j</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><munder><mo>⟶</mo><mrow><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>2</mn></msubsup></mrow></munder></mtd> <mtd><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mtd> <mtd><munder><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msubsup><mi>ϕ</mi> <mi>j</mi> <mn>2</mn></msubsup><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow></munder></mtd> <mtd><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (\phi^1_i \circ f_i^{-1})((x,i),v) = (\phi^1_j \circ f_j^{-1})( (x,j), g_{i j}(x)(v) ) \phantom{AAAA} \text{since:} \phantom{AAAA} \array{ && E_1 \\ & {}^{\mathllap{\phi^1_i \circ f_i^{-1}}}\nearrow & \uparrow^{\mathrlap{f^{-1}}} & \nwarrow^{\mathrlap{ \phi^1_j \circ f_j^{-1} }} \\ U_i \times k^n &\underset{\phi^2_i}{\longrightarrow}& (E_2)\vert_{U_i \cap U_i} &\underset{(\phi^2_j)^{-1}}{\longrightarrow}& U_i \times k^n } \,. </annotation></semantics></math></div> <p>Therefore by the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_364' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>E_2</annotation></semantics></math>, these functions <a class='existingWikiWord' href='/nlab/show/diff/extension'>extend</a> to a unique continuous function <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_365' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>E_2 \to E_1</annotation></semantics></math> such that the following <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>diagram commutes</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_366' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>i</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mn>1</mn></msubsup><mo>∘</mo><msubsup><mi>f</mi> <mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>E</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width='0'><mrow><mo>∃</mo><mo>!</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><msub><mi>E</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ \underset{i \in i}{\sqcup} U_i \times k^n &\overset{( \phi^1_i \circ f_i^{-1} )_{i \in I}}{\longrightarrow}& E_1 \\ \downarrow & \nearrow_{\mathrlap{\exists !}} \\ E_2 } \,. </annotation></semantics></math></div> <p>This unique function is clearly <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_367' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>f^{-1}</annotation></semantics></math> (by pointwise inspection) and therefore <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_368' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>f^{-1}</annotation></semantics></math> is continuous.</p> </div> <div class='num_example' id='FiberwiseLinearlyIndependentSectionsTrivialize'> <h6 id='example_14'>Example</h6> <p><strong>(fiberwise linearly independent sections trivialize a vector bundle)</strong></p> <p>If a topological vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_369' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/rank'>rank</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_370' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> admits <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_371' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/section'>sections</a> (example <a class='maruku-ref' href='#VectorBundleSections'>2</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_372' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>σ</mi> <mi>k</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⟶</mo><mi>E</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>k</mi><mo>∈</mo><mo stretchy='false'>{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy='false'>}</mo></mrow></msub></mrow><annotation encoding='application/x-tex'> \{\sigma_k \;\colon\; X \longrightarrow E\}_{k \in \{1, \cdots, n\}} </annotation></semantics></math></div> <p>that are linearly independent at each point <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_373' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_374' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is trivializable (example <a class='maruku-ref' href='#TrivialTopologicalVectorBundle'>1</a>). In fact, with the sections regarded as vector bundle homomorphisms out of the trivial vector bundle of rank <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_375' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> (according to example <a class='maruku-ref' href='#VectorBundleSections'>2</a>), these sections <em>are</em> the trivialization</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_376' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>σ</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>σ</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>E</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (\sigma_1, \cdots, \sigma_n) \;\colon\; (X \times k^n) \overset{\simeq}{\longrightarrow} E \,. </annotation></semantics></math></div> <p>This is because their linear independence at each point means precisely that this morphism of vector bundles is a fiber-wise linear isomorphsm and therefore an isomorphism of vector bundles by lemma <a class='maruku-ref' href='#FiberwiseIsoisIsomorphismOfVectorBundles'>1</a>.</p> </div> <h3 id='DirectSummandBundles'>Direct summand bundles</h3> <p>We discuss properties of the <a class='existingWikiWord' href='/nlab/show/diff/direct+sum+of+vector+bundles'>direct sum of vector bundles</a> for topological vector bundles.</p> <div class='num_prop' id='TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands'> <h6 id='proposition_4'>Proposition</h6> <p><strong>(sub-bundles over <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact spaces</a> are <a class='existingWikiWord' href='/nlab/show/diff/direct+sum+of+vector+bundles'>direct summands</a>)</strong></p> <p>Let</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_377' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+space'>paracompact Hausdorff space</a>,</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_378' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a> (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>).</p> </li> </ol> <p>Then every topological vector sub-bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_379' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mo>↪</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>E_1 \hookrightarrow E</annotation></semantics></math> (example <a class='maruku-ref' href='#TopologicalVetorSubbundle'>3</a>) is a direct vector bundle summand, in that there exists another vector sub-bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_380' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub><mo>↪</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>E_2 \hookrightarrow E</annotation></semantics></math> (example <a class='maruku-ref' href='#TopologicalVetorSubbundle'>3</a>) such that their <a class='existingWikiWord' href='/nlab/show/diff/direct+sum+of+vector+bundles'>direct sum of vector bundles</a> is <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_381' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_382' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mo>⊕</mo><msub><mi>E</mi> <mn>2</mn></msub><mo>≃</mo><mi>E</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> E_1 \oplus E_2 \simeq E \,. </annotation></semantics></math></div></div> <p>(<a href='#Hatcher'>e.g. Hatcher, prop. 1.3</a>)</p> <div class='proof'> <h6 id='proof_7'>Proof</h6> <p>Since <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_383' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is assumed to be paracompact Hausdorff, there exists an <a class='existingWikiWord' href='/nlab/show/diff/inner+product+on+vector+bundles'>inner product on vector bundles</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_384' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>⟨</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>⟩</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><msub><mo>⊕</mo> <mi>X</mi></msub><mi>E</mi><mo>⟶</mo><mi>X</mi><mo>×</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'> \langle -,-\rangle \;\colon\; E \oplus_X E \longrightarrow X \times k </annotation></semantics></math></div> <p>(by <a href='inner+product+of+vector+bundles#ExistenceOfInnerProductOfTopologicalVectorBundlesOverParacompactHausdorffSpaces'>this prop.</a>). This defines at each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_385' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/orthogonality'>orthogonal complement</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_386' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>E</mi><msub><mo>′</mo> <mi>x</mi></msub><msup><mo stretchy='false'>)</mo> <mo>⊥</mo></msup><mo>⊂</mo><msub><mi>E</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>(E'_x)^\perp \subset E_x</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_387' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><msub><mo>′</mo> <mi>x</mi></msub><mo>↪</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>E'_x \hookrightarrow E</annotation></semantics></math>. The <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> of these orthogonal complements is readily checked to be a <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_388' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>E</mi><mo>′</mo><msup><mo stretchy='false'>)</mo> <mo>⊥</mo></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>(E')^\perp \to X</annotation></semantics></math>. Hence by construction we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_389' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mi>E</mi><mo>′</mo><msub><mo>⊕</mo> <mi>X</mi></msub><mo stretchy='false'>(</mo><mi>E</mi><mo>′</mo><msup><mo stretchy='false'>)</mo> <mo>⊥</mo></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> E \;\simeq\; E' \oplus_X (E')^\perp \,. </annotation></semantics></math></div></div> <div class='num_prop' id='TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle'> <h6 id='proposition_5'>Proposition</h6> <p><strong>(vector bundles over a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a> are <a class='existingWikiWord' href='/nlab/show/diff/direct+sum+of+vector+bundles'>direct summands</a> of a <a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial vector bundle</a>)</strong></p> <p>Let</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_390' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_391' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a> (def. <a class='maruku-ref' href='#TopologicalVectorBundle'>2</a>).</p> </li> </ol> <p>Then there exists another topological vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_392' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>E</mi><mo stretchy='false'>˜</mo></mover><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\tilde E \to X</annotation></semantics></math> such that the <a class='existingWikiWord' href='/nlab/show/diff/direct+sum+of+vector+bundles'>direct sum of vector bundles</a> of the two is <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphic</a> to a <a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial vector bundle</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_393' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>X \times k^n</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_394' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>⊕</mo><mover><mi>E</mi><mo stretchy='false'>˜</mo></mover><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> E \oplus \tilde E \;\simeq\; X \times k^n \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_8'>Proof</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_395' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_396' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> over which <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_397' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>local trivialization</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_398' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mo>≃</mo></mover><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>}</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \left\{ \phi_i \;\colon\; U_i \times k^n \overset{\simeq}{\longrightarrow} E\vert_{U_i} \right\}_{i \in I} \,. </annotation></semantics></math></div> <p>By compactness of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_399' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, there is a <a class='existingWikiWord' href='/nlab/show/diff/finite+cover'>finite sub-cover</a>, hence a <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite set</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_400' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>J \subset I</annotation></semantics></math> such tat</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_401' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \{U_i \subset X\}_{i \in J \subset I} </annotation></semantics></math></div> <p>is still an open cover over which <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_402' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> trivializes.</p> <p>Since <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a> there exists a <a class='existingWikiWord' href='/nlab/show/diff/partition+of+unity'>partition of unity</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_403' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>f</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>→</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>}</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \left\{ f_i \;\colon\; X \to [0,1] \right\}_{i \in J} </annotation></semantics></math></div> <p>with <a class='existingWikiWord' href='/nlab/show/diff/support'>support</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_404' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>supp</mi><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>⊂</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>supp(f_i) \subset U_i</annotation></semantics></math>. Hence the functions</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_405' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mphantom><mi>AAAA</mi></mphantom></mover></mtd> <mtd><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mi>v</mi></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>⋅</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ E\vert_{U_i} &\overset{\phantom{AAAA}}{\longrightarrow}& U_i \times k^n \\ v &\overset{\phantom{AAA}}{\mapsto}& f_i(x) \cdot \phi_i^{-1}(v) } </annotation></semantics></math></div> <p>extend by 0 to vector bundle homomorphism of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_406' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>⋅</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><mo>⟶</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f_i \cdot \phi^{-1}_i \;\colon\; E \longrightarrow X \times k^n \,. </annotation></semantics></math></div> <p>The finite pointwise <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a> of these yields a vector bundle homomorphism of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_407' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>⊕</mo><mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><mo>⟶</mo><mi>X</mi><mo>×</mo><mrow><mo>(</mo><munder><mo>⊕</mo><mrow><mi>i</mi><mo>∈</mo><mi>J</mi></mrow></munder><msup><mi>k</mi> <mi>n</mi></msup><mo>)</mo></mrow><mo>≃</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><mi>n</mi><mover><mrow><mo stretchy='false'>|</mo><mi>J</mi><mo stretchy='false'>|</mo></mrow><mo>˙</mo></mover></mrow></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \underset{i \in J}{\oplus} f_i \cdot \phi_i \;\colon\; E \longrightarrow X \times \left( \underset{i \in J}{\oplus} k^n \right) \simeq X \times k^{n \dot {\vert J\vert}} \,. </annotation></semantics></math></div> <p>Observe that, as opposed to the single <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_408' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>⋅</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>f_i \cdot \phi^{-1}_i</annotation></semantics></math>, this is a fiber-wise injective, because at each point at least one of the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_409' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>f_i</annotation></semantics></math> is non-vanishing. Hence this is an injection of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_410' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> into a trivial vector bundle.</p> <p>With this the statement follows by prop. <a class='maruku-ref' href='#TopologicalSubBundlesOverParacompactHausdorffSpacesAreDirectSummands'>4</a>.</p> </div> <div class='num_remark'> <h6 id='remark_6'>Remark</h6> <p>Prop. <a class='maruku-ref' href='#TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle'>5</a> is key in the analysis of <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a> groups on <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff spaces</a>. See <a href='topological+K-theory#DirectSumHasInverseUpToTrivialBundle'>there</a> for more.</p> </div> <h3 id='ConcordanceOfTopolgicslVectorBundles'>Concordance</h3> <p>We discuss that every <a class='existingWikiWord' href='/nlab/show/diff/concordance'>concordance</a> of topological vector bundles over a <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact topological space</a> makes the restriction of the vector bundle over the endpoints of the interval isomorphic (prop. <a class='maruku-ref' href='#ConcondanceOfTopologicalVectorBundles'>6</a> below). In particular this implies tht the <a class='existingWikiWord' href='/nlab/show/diff/pullback+bundle'>pullbacks of vector bundles</a> along two <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopic</a> <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> are <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphic</a> (corollary <a class='maruku-ref' href='#PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic'>1</a> below).</p> <p>This result is apparently due to Steenrod, see Theorem 11.4 in <a href='#Steenrod'>Steenrod</a>. The proof below follows <a href='#Hatcher'>Hatcher, theorem 1.6</a>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_411' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> write <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_412' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X \times I</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> with the <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_413' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[0,1]</annotation></semantics></math> equipped with its <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>.</p> <p>Write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_414' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mover><mo>⟵</mo><mrow><msub><mi>p</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow></msub></mrow></mover><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> X \overset{p_X}{\longleftarrow} X \times [0,1] \overset{p_{[0,1]}}{\longrightarrow} [0,1] </annotation></semantics></math></div> <p>for the two continuous <a class='existingWikiWord' href='/nlab/show/diff/projection'>projections</a> out of the product space.</p> <div class='num_lemma' id='TrivilizationOfVectorBundleOverProductSpaceWithInterval'> <h6 id='lemma_2'>Lemma</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_415' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, then a vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_416' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>E \to X \times [0,1]</annotation></semantics></math> is trivializable (example <a class='maruku-ref' href='#TrivialTopologicalVectorBundle'>1</a>) if its restrictions to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_417' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>X \times [0,1/2]</annotation></semantics></math> and to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_418' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>X \times [1/2,1]</annotation></semantics></math> are trivializable.</p> </div> <div class='num_lemma' id='CoverForProductSpaceWithIntrval'> <h6 id='lemma_3'>Lemma</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_419' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, then for every topological vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_420' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>E \to X \times I</annotation></semantics></math> there exists an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_421' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_422' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> such that the vector bundle trivializes over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_423' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>⊂</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>U_i \times [0,1] \subset X \times [0,1]</annotation></semantics></math>, for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_424' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i \in I</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_9'>Proof</h6> <p>By <a class='existingWikiWord' href='/nlab/show/diff/local+trivialization'>local trvializability</a> of the vector bundle, there exists an open cover <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_425' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>j</mi></msub><mo>⊂</mo><mi>X</mi><mo>×</mo><mi>I</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{V_j \subset X \times I\}_{j \in J}</annotation></semantics></math> over which the bundle trivializes. For each point <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_426' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> this induces a cover of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_427' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\{x\} \times [0,1]</annotation></semantics></math>. This is a <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact topological space</a> (for instance by the <a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a>) and hence there exists a <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_428' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>J</mi> <mi>x</mi></msub><mo>⊂</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>J_x \subset I</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_429' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><mo>×</mo><mi>I</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><msub><mi>J</mi> <mi>x</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\{V_i \subset X \times I\}_{i \in J_x}</annotation></semantics></math> still covers <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_430' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\{x\} \times [0,1]</annotation></semantics></math>.</p> <p>By finiteness of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_431' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>J</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>J_x</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_432' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>≔</mo><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><msub><mi>J</mi> <mi>x</mi></msub></mrow></munder><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy='false'>(</mo><msub><mi>V</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> U_x \coloneqq \underset{i \in J_x}{\cap} p_X(V_i) </annotation></semantics></math></div> <p>is an open neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_433' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_434' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. Moreover</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_435' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>p</mi> <mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow></msub><mo stretchy='false'>(</mo><msub><mi>V</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>⊂</mo><mi>I</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><msub><mi>J</mi> <mi>x</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'> \{ p_{[0,1]}(V_i) \subset I \}_{i \in J_x} </annotation></semantics></math></div> <p>is an open cover of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_436' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[0,1]</annotation></semantics></math> such that the given vector bundle trivializes over each element of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_437' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>x</mi></msub><mo>×</mo><msub><mi>p</mi> <mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow></msub><mo stretchy='false'>(</mo><msub><mi>V</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><msub><mi>J</mi> <mi>x</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_x \times p_{[0,1]}(V_i)\}_{i \in J_x}</annotation></semantics></math>.</p> <p>By the nature of the Euclidean <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a> each <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_438' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[0,1]</annotation></semantics></math> is a union of intervals. So we may pass to a <a class='existingWikiWord' href='/nlab/show/diff/refinement'>refinement</a> of this cover of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_439' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[0,1]</annotation></semantics></math> such that each element is a single interval. Again by compactness of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_440' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[0,1]</annotation></semantics></math>, this refinement has a finite subcover</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_441' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>W</mi> <mrow><mi>x</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>⊂</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><msub><mo stretchy='false'>}</mo> <mrow><mi>k</mi><mo>∈</mo><msub><mi>K</mi> <mi>x</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'> \{W_{x,k} \subset [0,1]\}_{k \in K_x} </annotation></semantics></math></div> <p>each element of which is an <a class='existingWikiWord' href='/nlab/show/diff/interval'>interval</a>. Since this is a finite cover, we may find numbers <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_442' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>0</mn><mo>=</mo><msub><mi>t</mi> <mn>0</mn></msub><mo><</mo><msub><mi>t</mi> <mn>1</mn></msub><mo><</mo><msub><mi>t</mi> <mn>2</mn></msub><mo><</mo><mi>⋯</mi><mo><</mo><msub><mi>t</mi> <mrow><msub><mi>n</mi> <mi>x</mi></msub></mrow></msub><mo>=</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{0 = t_0 \lt t_1 \lt t_2 \lt \cdots \lt t_{n_x} = 1\}</annotation></semantics></math> such that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_443' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mo stretchy='false'>[</mo><msub><mi>t</mi> <mi>k</mi></msub><mo>,</mo><msub><mi>t</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>]</mo><mo>⊂</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><msub><mo stretchy='false'>}</mo> <mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo><</mo><msub><mi>n</mi> <mi>x</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'> \{ [t_k, t_{k+1}] \subset [0,1] \}_{0 \leq k \lt n_x} </annotation></semantics></math></div> <p>is a cover of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_444' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[0,1]</annotation></semantics></math>, and such that the given vector bundle still trivializes over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_445' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mi>x</mi></msub><mo>×</mo><mo stretchy='false'>[</mo><msub><mi>t</mi> <mi>k</mi></msub><mo>,</mo><msub><mi>t</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>V_x \times [t_k, t_{k+1}]</annotation></semantics></math> for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_446' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo><</mo><msub><mi>n</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>0 \leq k \lt n_x</annotation></semantics></math>.</p> <p>By lemma <a class='maruku-ref' href='#TrivilizationOfVectorBundleOverProductSpaceWithInterval'>2</a> this implies that the vector bundle in fact trivializes over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_447' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>U_x \times [0,1]</annotation></semantics></math>.</p> <p>Applying this procedure for all points <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_448' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> yields a cover</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_449' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \{ U_x \subset X \}_{x \in X} </annotation></semantics></math></div> <p>with the required property.</p> </div> <div class='num_prop' id='ConcondanceOfTopologicalVectorBundles'> <h6 id='proposition_6'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/concordance'>concordance</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundles</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_450' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+space'>paracompact Hausdorff space</a>. If <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_451' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>E \to X \times [0,1]</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a> over the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_452' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a> (hence a <em><a class='existingWikiWord' href='/nlab/show/diff/concordance'>concordance</a></em> of topological vector bundles on <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_453' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>), then the two endpoint-restrictions</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_454' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo></mrow></msub><mphantom><mi>AA</mi></mphantom><mtext>and</mtext><mphantom><mi>AA</mi></mphantom><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow></msub></mrow><annotation encoding='application/x-tex'> E|_{X \times \{0\}} \phantom{AA} \text{and} \phantom{AA} E|_{X \times \{1\}} </annotation></semantics></math></div> <p>are <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphic</a> vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_455' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_10'>Proof</h6> <p>By lemma <a class='maruku-ref' href='#CoverForProductSpaceWithIntrval'>3</a> there exists an open cover <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_456' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_457' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> such that the vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_458' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> trivializes over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_459' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>U_i \times [0,1]</annotation></semantics></math> for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_460' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i \in I</annotation></semantics></math>. By <a href='paracompact+topological+space#CountableCoverOfUnionsofOpenSubsetsInsideGivenCover'>this lemma</a> there exists a <a class='existingWikiWord' href='/nlab/show/diff/countable+cover'>countable cover</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_461' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>n</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \{V_n \subset X\}_{n \in \mathbb{N}} </annotation></semantics></math></div> <p>such that each element is a <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> of open subsets that each are contained in one of the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_462' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math>. This means that the vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_463' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> still trivializes over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_464' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mi>n</mi></msub><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>V_n \times [0,1]</annotation></semantics></math>, for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_465' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>.</p> <p>Moreover, since <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a>, there exists a <a class='existingWikiWord' href='/nlab/show/diff/partition+of+unity'>partition of unity</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_466' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>f</mi> <mi>n</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>}</mo></mrow> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\left\{f_n \colon X \to [0,1] \right\}_{n \in \mathbb{N}}</annotation></semantics></math> subordinate to this countable cover.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_467' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> define</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_468' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ψ</mi> <mi>n</mi></msub><mo>≔</mo><munderover><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow><mi>n</mi></munderover><msub><mi>f</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'> \psi_n \coloneqq \underoverset{k = 0}{n}{\sum} f_n </annotation></semantics></math></div> <p>(so <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_469' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ψ</mi> <mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\psi_0 = 0</annotation></semantics></math> and by local finiteness there is for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_470' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> an <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_471' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>n</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>n_x</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_472' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ψ</mi> <mrow><mi>n</mi><mo>></mo><msub><mi>n</mi> <mi>x</mi></msub></mrow></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\psi_{n \gt n_x} = 1</annotation></semantics></math>.)</p> <p>Now write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_473' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><mo>≔</mo><mi>graph</mi><mo stretchy='false'>(</mo><msub><mi>ψ</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo>⊂</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'> X_n \coloneqq graph( \psi_n ) \subset X \times [0,1] </annotation></semantics></math></div> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/graph'>graph</a> of the function <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_474' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ψ</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\psi_n</annotation></semantics></math> equipped with its <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, and write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_475' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub><mo>≔</mo><msubsup><mi>ψ</mi> <mi>n</mi> <mo>*</mo></msubsup><mi>E</mi></mrow><annotation encoding='application/x-tex'> E_n \coloneqq \psi_n^\ast E </annotation></semantics></math></div> <p>for the restriction of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_476' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> to that subspace</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_477' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>E</mi> <mi>n</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mi>n</mi></msub><mo>=</mo><mi>graph</mi><mo stretchy='false'>(</mo><msub><mi>ψ</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ E_n &\longrightarrow& E \\ \downarrow && \downarrow \\ X_n = graph(\psi_n) &\hookrightarrow& X } </annotation></semantics></math></div> <p>Observe that the projection functions</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_478' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>p</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>n</mi></mrow></msub><mo lspace='verythinmathspace'>:</mo></mtd> <mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd><mover><mo>⟶</mo><mrow /></mover></mtd> <mtd><msub><mi>X</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd /> <mtd><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><msub><mi>ψ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AA</mi></mphantom></mover></mtd> <mtd><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><msub><mi>ψ</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><msub><mi>ψ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>−</mo><msub><mi>f</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ p_{n+1,n} \colon & X_{n+1} &\overset{}{\longrightarrow}& X_n \\ & (x,\psi_{n+1}(x)) &\overset{\phantom{AA}}{\mapsto}& (x, \psi_n(x)) = (x, \psi_{n+1}(x) - f_{n+1}(x)) } </annotation></semantics></math></div> <p>are <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a>: By the nature of the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topology</a> and the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> it is sufficient to check for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_479' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_480' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>V \subset \mathbb{R}</annotation></semantics></math> open subsets, that every point <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_481' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>c</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x,c)</annotation></semantics></math> in the preimage <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_482' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>p</mi> <mi>n</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>p_n^{-1}( U \times V ) \subset X \times [0,1]</annotation></semantics></math> is contained in an open subset of the form <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_483' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>×</mo><msub><mi>V</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>U_x \times V_x \subset X \times [0,1]</annotation></semantics></math> such that every point of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_484' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>X_{n+1}</annotation></semantics></math> that is also in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_485' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>×</mo><msub><mi>V</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>U_x \times V_x</annotation></semantics></math> is still mapped to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_486' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>×</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>U \times V</annotation></semantics></math>. Such an open subset is <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_487' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><mi>U</mi><mo>∩</mo><msubsup><mi>ψ</mi> <mi>n</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\left( U \cap \psi_n^{-1}(V) \right) \times [0,1]</annotation></semantics></math>.</p> <p>Also observe that the composites</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_488' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub><mo>⟶</mo><msub><mi>X</mi> <mi>n</mi></msub><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msub></mrow></mover><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'> E_n \longrightarrow X_n \overset{p_{n,0}}{\longrightarrow} X_0 = 0 </annotation></semantics></math></div> <p>make each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_489' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>E_n</annotation></semantics></math> a vector bundle over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_490' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>: To see local trivializability over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_491' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> choose a local trivialization of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_492' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> over some open cover <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_493' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> and observe that then <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_494' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>E_n</annotation></semantics></math> is trivial over the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>fiber product</a><math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_495' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub><msub><mo>×</mo> <mi>X</mi></msub><msub><mi>U</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>X_n \times_X U_n</annotation></semantics></math> and hence over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_496' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>U_n</annotation></semantics></math>.</p> <p>Now by the pullback definition of the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_497' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>E_n</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/pasting+law+for+pullbacks'>pasting law</a> says that for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_498' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> we have a pullback square of vector bundles of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_499' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd /> <mtd><mover><mo>⟶</mo><mrow><msub><mi>h</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd /> <mtd><msub><mi>E</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>(</mo><mi>pb</mi><mo stretchy='false'>)</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd> <mtd /> <mtd><mo>⟶</mo></mtd> <mtd /> <mtd><msub><mi>X</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd /> <mtd><mo>↘</mo></mtd> <mtd /> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ E_{n+1} && \overset{h_n}{\longrightarrow} && E_n \\ \downarrow && (pb) && \downarrow \\ X_{n+1} && \longrightarrow && X_n \\ & \searrow && \swarrow \\ && X } \,. </annotation></semantics></math></div> <p>By the nature of pullbacks, the top horizontal function <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_500' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>h_n</annotation></semantics></math> in this diagram is on each fiber a linear isomorphism. Therefore prop. <a class='maruku-ref' href='#FiberwiseIsoisIsomorphismOfVectorBundles'>1</a> implies that each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_501' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>h_n</annotation></semantics></math> is in fact an isomorphism of vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_502' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> <p>By local finiteness, each point <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_503' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> has a neighbourhood <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_504' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>U_x</annotation></semantics></math> such that only a finite number <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_505' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>n</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>n_x</annotation></semantics></math> of these <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_506' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>h_n</annotation></semantics></math> are non-trivial, and so it makes sense to consider the infinite composition</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_507' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi><mo>≔</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>∘</mo><msub><mi>h</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>h</mi> <mn>3</mn></msub><mo>∘</mo><mi>⋯</mi></mrow><annotation encoding='application/x-tex'> h \coloneqq h_1 \circ h_2 \circ h_3 \circ \cdots </annotation></semantics></math></div> <p>understood to be on each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_508' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>U_x</annotation></semantics></math> the finite composite</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_509' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≔</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>∘</mo><mi>⋯</mi><mo>∘</mo><msub><mi>h</mi> <mrow><msub><mi>n</mi> <mi>x</mi></msub></mrow></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> h(x) \coloneqq h_1 \circ \cdots \circ h_{n_x} \,. </annotation></semantics></math></div> <p>Since all the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_510' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>h_k</annotation></semantics></math> are vector bundle isomorphisms, so are all their composites. Thus <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_511' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi></mrow><annotation encoding='application/x-tex'>h</annotation></semantics></math> is an isomorphism of the required form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_512' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo></mrow></msub><mover><mo>⟶</mo><mo>≃</mo></mover><mi>E</mi><msub><mo stretchy='false'>|</mo> <mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> h \;\colon\; E|_{X \times \{0\}} \overset{\simeq}{\longrightarrow} E|_{X \times \{1\}} \,. </annotation></semantics></math></div></div> <div class='num_cor' id='PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic'> <h6 id='corollary'>Corollary</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_513' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+space'>paracompact Hausdorff space</a>, let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_514' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>E \to Y</annotation></semantics></math> be a topological <a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a>, let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_515' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f,g \colon X \to Y</annotation></semantics></math> be two <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a>, and let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_516' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mo>→</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>\eta \colon f \to g</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a> between them. Then there is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> of vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_517' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> between the <a class='existingWikiWord' href='/nlab/show/diff/pullback+bundle'>pullback of vector bundles</a> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_518' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_519' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and along <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_520' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math>, respectively:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_521' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>E</mi><mo>≃</mo><msup><mi>g</mi> <mo>*</mo></msup><mi>E</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f^\ast E \simeq g^\ast E \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_11'>Proof</h6> <p>By definition, the <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_522' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi></mrow><annotation encoding='application/x-tex'>\eta</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_523' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>⟶</mo><mi>Y</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \eta \;\colon\; X \times [0,1] \longrightarrow Y \,. </annotation></semantics></math></div> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_524' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>∈</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>t \in [0,1]</annotation></semantics></math> write <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_525' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>t</mi></msub></mrow><annotation encoding='application/x-tex'>i_t</annotation></semantics></math> for the continuous function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_526' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>i</mi> <mi>t</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAAA</mi></mphantom></mover></mtd> <mtd><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ X &\overset{\phantom{AA}i_t\phantom{AA}}{\longrightarrow}& X \times [0,1] \\ x &\overset{\phantom{AAAA}}{\mapsto}& (x,t) } \,. </annotation></semantics></math></div> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/pasting+law+for+pullbacks'>pasting law</a> for pullbacks we have that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_527' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>E</mi><mo>=</mo><mo stretchy='false'>(</mo><mi>η</mi><mo>∘</mo><msub><mi>i</mi> <mn>0</mn></msub><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><mi>E</mi><mo>≃</mo><msubsup><mi>i</mi> <mn>0</mn> <mo>*</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>η</mi> <mo>*</mo></msup><mi>E</mi><mo stretchy='false'>)</mo><mo>≃</mo><mo stretchy='false'>(</mo><msup><mi>η</mi> <mo>*</mo></msup><mi>E</mi><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo></mrow></msub></mrow><annotation encoding='application/x-tex'> f^\ast E = (\eta \circ i_0)^\ast E \simeq i_0^\ast (\eta^\ast E) \simeq (\eta^\ast E)|_{X \times \{0\}} </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_528' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>g</mi> <mo>*</mo></msup><mi>E</mi><mo>=</mo><mo stretchy='false'>(</mo><mi>η</mi><mo>∘</mo><msub><mi>i</mi> <mn>1</mn></msub><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><mi>E</mi><mo>≃</mo><msubsup><mi>i</mi> <mn>1</mn> <mo>*</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>η</mi> <mo>*</mo></msup><mi>E</mi><mo stretchy='false'>)</mo><mo>≃</mo><mo stretchy='false'>(</mo><msup><mi>η</mi> <mo>*</mo></msup><mi>E</mi><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow></msub></mrow><annotation encoding='application/x-tex'> g^\ast E = (\eta \circ i_1)^\ast E \simeq i_1^\ast (\eta^\ast E) \simeq (\eta^\ast E)|_{X \times \{1\}} </annotation></semantics></math></div> <p>With this the statement follows by prop. <a class='maruku-ref' href='#ConcondanceOfTopologicalVectorBundles'>6</a>.</p> </div> <div class='num_example' id='HomotopyInvarianceOfIsomorphismClassesOfVectorBundles'> <h6 id='example_15'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/homotopy+invariance'>homotopy invariance</a> of isomorphism classes of vector bundles)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_529' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_530' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> be <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+space'>paracompact Hausdorff spaces</a> and let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_531' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'> f \;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>be a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> which is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>. Then pullback along <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_532' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> constitutes a <a class='existingWikiWord' href='/nlab/show/diff/bijection'>bijection</a> on sets of isomorphism classes of topological vector bundles:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_533' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>Vect</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo><msub><mo stretchy='false'>/</mo> <mo>∼</mo></msub><mover><mo>⟶</mo><mo>≃</mo></mover><mi>Vect</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><msub><mo stretchy='false'>/</mo> <mo>∼</mo></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f^\ast \;\colon\; Vect(Y)/_\sim \overset{\simeq}{\longrightarrow} Vect(X)/_\sim \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_12'>Proof</h6> <p>By definition of homotopy equivalence, there is a continuous function <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_534' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>Y</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>g \colon Y \longrightarrow X</annotation></semantics></math> and <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopies</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_535' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>∘</mo><mi>f</mi><mo>⇒</mo><mi>id</mi><mphantom><mi>AAAA</mi></mphantom><mi>f</mi><mo>∘</mo><mi>g</mi><mo>⇒</mo><mi>id</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> g \circ f \Rightarrow id \phantom{AAAA} f \circ g \Rightarrow id \,. </annotation></semantics></math></div> <p>Hence corollary <a class='maruku-ref' href='#PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic'>1</a> implies that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_536' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>∘</mo><msup><mi>g</mi> <mo>*</mo></msup><mo>=</mo><mo stretchy='false'>(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><mo>=</mo><mi>id</mi><mphantom><mi>AAAAA</mi></mphantom><msup><mi>g</mi> <mo>*</mo></msup><mo>∘</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>=</mo><mo stretchy='false'>(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><mo>=</mo><mi>id</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f^\ast \circ g^\ast = (g \circ f)^\ast = id \phantom{AAAAA} g^\ast \circ f^\ast = (f \circ g)^\ast = id \,. </annotation></semantics></math></div> <p>This mean that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_537' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>g^\ast</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/inverse+function'>inverse function</a> to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_538' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^\ast</annotation></semantics></math>, and hence both are bijections.</p> </div> <div class='num_example' id='TopologicalVectorBundleOverContractibleSpaceIsTrivializable'> <h6 id='example_16'>Example</h6> <p><strong>(topological vector bundle on contractible topological space is trivializable)</strong></p> <p>If <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_539' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible topological space</a>, then every topological vector bundle over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_540' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is isomorphic to a <a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial vector bundle</a>.</p> </div> <div class='proof'> <h6 id='proof_13'>Proof</h6> <p>That <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_541' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is contractible means by definition that there is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_542' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mpadded><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded><mo stretchy='false'>↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width='0'><mi>id</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd /></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ X &\longrightarrow& \ast \\ \mathllap{i_0}\downarrow & & \downarrow \\ X \times [0,1] &\overset{\eta}{\longrightarrow}& X \\ \mathllap{i_1}\uparrow & \nearrow_{\mathrlap{id}} \\ X & } \,. </annotation></semantics></math></div> <p>By cor <a class='maruku-ref' href='#PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic'>1</a> it follows that for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_543' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> any topological vector bundle that there is an isomorphism between <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_544' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>id</mi> <mo>*</mo></msup><mi>E</mi><mo>=</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>id^\ast E = E</annotation></semantics></math> and the result of first restricting the bundle to the point, and then forming the <a class='existingWikiWord' href='/nlab/show/diff/pullback+bundle'>pullback bundle</a> along <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_545' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>X \to \ast</annotation></semantics></math>. But the latter operation precisely produces the <a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial vector bundles</a> over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_546' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> </div> <h3 id='OverClosedSubspaces'>Over closed subspaces</h3> <p>We discuss the behavour of vector bundles with respect to <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subspaces</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_547' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A \subset X</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff spaces</a>.</p> <div class='num_lemma' id='IsomorphismOfVectorBundlesOnClosedSubsetOfCompactHausdorffSpaceExtendsToOpenNeighbourhoods'> <h6 id='lemma_4'>Lemma</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> of vector bundles on <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a> of <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff spaces</a> <a class='existingWikiWord' href='/nlab/show/diff/extension'>extends</a> to <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_548' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>∈</mo><mo stretchy='false'>{</mo><mi>ℝ</mi><mo>,</mo><mi>ℂ</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>k \in \{\mathbb{R}, \mathbb{C}\}</annotation></semantics></math>, let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_549' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a> and let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_550' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A \subset X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>. Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_551' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'>E_i \overset{p_i}{\to} X</annotation></semantics></math> be two topological vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_552' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_553' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy='false'>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>i \in \{1,2\}</annotation></semantics></math>.</p> <p>If there exists an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_554' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'> E_1\vert_A \overset{\simeq}{\longrightarrow} E_2\vert_A </annotation></semantics></math></div> <p>of the restricted vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_555' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, then there also exists an <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_556' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_557' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>A \subset U</annotation></semantics></math> such that there is also an isomorphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_558' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>|</mo> <mi>U</mi></msub><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mi>U</mi></msub></mrow><annotation encoding='application/x-tex'> E_1\vert_U \overset{\simeq}{\longrightarrow} E_2\vert_U </annotation></semantics></math></div> <p>of the vector bundles restricted to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_559' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_14'>Proof</h6> <p>A bundle isomorphism <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_560' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub><mo>≃</mo><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>E_1\vert_A \simeq E_2\vert_A</annotation></semantics></math> is equivalently a trivializing section (example <a class='maruku-ref' href='#FiberwiseLinearlyIndependentSectionsTrivialize'>14</a>) of the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+vector+bundles'>tensor product of vector bundles</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_561' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>(E_1\vert_A)^\ast \otimes_A E_2\vert_A</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_562' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>E_2\vert_A</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/dual+vector+bundle'>dual vector bundle</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_563' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>(E_2\vert_A)^\ast</annotation></semantics></math>. (by <a href='tensor+product+of+vector+bundles#FinitrRankBundleHomomorphismIsSectionOfTensorProductWithDual'>this prop.</a>).</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_564' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{V_i \subset X\}_{i \in I}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_565' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> over which this tensor product bundle trivializes with trivializations</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_566' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>{</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>ℝ</mi> <mrow><mo stretchy='false'>(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy='false'>)</mo></mrow></msup><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></munderover><mo stretchy='false'>(</mo><msubsup><mi>E</mi> <mn>1</mn> <mo>*</mo></msubsup><msub><mo>⊗</mo> <mi>X</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>}</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \left\{ V_i \times \mathbb{R}^{(n^2)} \underoverset{\simeq}{\phi_i}{\longrightarrow} (E_1^\ast \otimes_X E_2)\vert_{U_i} \right\} \,. </annotation></semantics></math></div> <p>Since <a class='existingWikiWord' href='/nlab/show/diff/compact+Hausdorff+spaces+are+normal'>compact Hausdorff spaces are normal</a>, the <a class='existingWikiWord' href='/nlab/show/diff/shrinking+lemma'>shrinking lemma</a> applies and gives a refinement of this by a cover <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_567' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> by <em>closed</em> subsets <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_568' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U_i \subset X</annotation></semantics></math>.</p> <p>Then a trivializing section <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_569' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>Γ</mi> <mi>A</mi></msub><mrow><mo>(</mo><mo stretchy='false'>(</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>\sigma \in \Gamma_A\left( (E_1\vert_A)^\ast \otimes_A E_2 \vert_A \right)</annotation></semantics></math> as above is on each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_570' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>U_i \cap A</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_571' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><mi>A</mi><mo>⟶</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \sigma_i \;\colon\; U_i \cap A \longrightarrow GL(n,k) </annotation></semantics></math></div> <p>to the <a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_572' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>⊂</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>GL(n,k) \subset Mat_{n \times n}(k)</annotation></semantics></math>, such that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_573' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><mi>A</mi></mrow></msub><mo>=</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>σ</mi> <mi>i</mi></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \sigma\vert_{U_i \cap A} = \phi_i \circ \sigma_i \,. </annotation></semantics></math></div> <p>Regarded as a function to the <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_574' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow><annotation encoding='application/x-tex'>n \times n</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/matrix'>matrices</a>, this is a set of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_575' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>n</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>n^2</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_576' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><msub><mi>σ</mi> <mi>i</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>a</mi><mi>b</mi></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>((\sigma_i)_{a b})</annotation></semantics></math></p> <p>Now since <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_577' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U_i \subset X</annotation></semantics></math> is closed by construction, and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_578' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A \subset X</annotation></semantics></math> is closed by assumption, also the intersections <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_579' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U_i \cap X</annotation></semantics></math> are closed. Since <a class='existingWikiWord' href='/nlab/show/diff/compact+Hausdorff+spaces+are+normal'>compact Hausdorff spaces are normal</a> the <a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a> therefore applies to these component functions and yields <a class='existingWikiWord' href='/nlab/show/diff/extension'>extensions</a> of each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_580' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\sigma_i</annotation></semantics></math> to a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_581' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>U</mi> <mi>i</mi></msub><mo>⟶</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \hat \sigma_i \;\colon\; U_i \longrightarrow Mat_{n \times n}(k) \,. </annotation></semantics></math></div> <p>Moreover, since compact Hausdorff spaces are evidently <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+space'>paracompact Hausdorff spaces</a>, and since <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a>, it follows that we find a <a class='existingWikiWord' href='/nlab/show/diff/partition+of+unity'>partition of unity</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_582' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>f</mi> <mi>i</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>ℝ</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{f_i \colon U_i \to \mathbb{R} \}_{i \in I}</annotation></semantics></math>.</p> <p>Consider then the functions <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_583' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>f_i \cdot \hat \sigma_i</annotation></semantics></math> given by pointwise multiplication and regarded, via extension by zero, as continuous functions on all of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_584' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_585' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover> <mi>i</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⟶</mo><mi>ℝ</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f_i \cdot \hat \sigma_i \;\colon\; X \longrightarrow \mathbb{R} \,. </annotation></semantics></math></div> <p>Summing these up yields a single section <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_586' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\hat \sigma</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_587' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>E</mi> <mn>1</mn> <mo>*</mo></msubsup><msub><mo>⊗</mo> <mi>X</mi></msub><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>E_1^\ast \otimes_X E_2</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_588' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover><mo>≔</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy='false'>(</mo><msubsup><mi>E</mi> <mn>1</mn> <mo>*</mo></msubsup><msub><mo>⊗</mo> <mi>X</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \hat \sigma \coloneqq \sum_{i \in I} \phi_i(f_i \cdot \hat \sigma_i) \in \Gamma_X(E_1^\ast \otimes_X E_2) \,, </annotation></semantics></math></div> <p>which by construction is an <a class='existingWikiWord' href='/nlab/show/diff/extension'>extension</a> of the original section, in that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_589' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub><mo>=</mo><mi>σ</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \hat \sigma\vert_A = \sigma \,. </annotation></semantics></math></div> <p>This is because for each <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_590' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>a \in A \subset X</annotation></semantics></math> we have, using the above definitions,</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_591' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>)</mo></mrow><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mo stretchy='false'>(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><msub><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><msub><mi>σ</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>⋅</mo><mo stretchy='false'>(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>∘</mo><msub><mi>σ</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>⋅</mo><mi>σ</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>f</mi> <mi>i</mi></msub><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><mo>⋅</mo><mi>σ</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mi>σ</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} \left(\underset{i \in I}{\sum} \phi_i(f_i \cdot \hat \sigma_i)\right)(a) & = \underset{i \in I}{\sum} (\phi_i (\hat \sigma_i(a))) \\ & = \underset{i \in I}{\sum} \phi_i( f_i(a) \sigma_i(a) ) \\ & = \underset{i \in I}{\sum} f_i(a) \cdot (\phi_i \circ \sigma_i)(a) \\ & = \underset{i \in I}{\sum} f_i(a) \cdot \sigma(a) \\ & = \left( \underset{i \in I}{\sum} f_i(a)\right) \cdot \sigma(a) \\ & = \sigma(a) \end{aligned} </annotation></semantics></math></div> <p>Here the last step uses the nature of the partition of unity.</p> <p>Now while <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_592' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\hat \sigma</annotation></semantics></math> is an extension of the section <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_593' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_594' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, it will in general not be a trivializing section on <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_595' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>But since the <a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_596' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>=</mo><msup><mi>det</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>k</mi><mo>∖</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>⊂</mo><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>GL(n,k) = det^{-1}(k \setminus \{0\}) \subset Mat_{n \times n}(k)</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a> of the <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_597' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Mat</mi> <mrow><mi>n</mi><mo>×</mo><mi>n</mi></mrow></msub><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>≃</mo><msup><mi>k</mi> <mrow><mo stretchy='false'>(</mo><msup><mi>n</mi> <mn>2</mn></msup><mo stretchy='false'>)</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>Mat_{n \times n}(k) \simeq k^{(n^2)}</annotation></semantics></math>, it follows that each point <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_598' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x \in A</annotation></semantics></math> has an open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_599' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U_x \subset X</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_600' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover><msub><mo stretchy='false'>|</mo> <mrow><msub><mi>U</mi> <mi>x</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\hat \sigma\vert_{U_x}</annotation></semantics></math> is still a trivializing section, namely choosing <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_601' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>x</mi></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i_x \in I</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_602' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mi>x</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>x \in U_{i_x}</annotation></semantics></math> set</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_603' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>≔</mo><mo stretchy='false'>(</mo><msub><mover><mi>σ</mi><mo stretchy='false'>^</mo></mover> <mrow><msub><mi>i</mi> <mi>x</mi></msub></mrow></msub><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>GL</mi><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> U_x \coloneqq (\hat \sigma_{i_x})^{-1}( GL(n,k) ) \,. </annotation></semantics></math></div> <p>The union of these</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_604' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>≔</mo><munder><mo>∪</mo><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow></munder><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'> U \coloneqq \underset{x \in A}{\cup} U_x </annotation></semantics></math></div> <p>is hence an open subset containing <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_605' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_606' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>E</mi> <mn>1</mn> <mo>*</mo></msubsup><msub><mo>⊗</mo> <mi>X</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>|</mo> <mi>U</mi></msub></mrow><annotation encoding='application/x-tex'>(E_1^\ast \otimes_X E_2)\vert_U</annotation></semantics></math> has a trivializing section, extending <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_607' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math>, hence such that there is an isomorphism <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_608' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><msub><mo stretchy='false'>|</mo> <mi>U</mi></msub><mo>≃</mo><msub><mi>E</mi> <mn>2</mn></msub><msub><mo stretchy='false'>|</mo> <mi>U</mi></msub></mrow><annotation encoding='application/x-tex'>E_1\vert_U \simeq E_2 \vert_U</annotation></semantics></math> extending the original isomorphism on <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_609' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> </div> <p>As a consequence:</p> <div class='num_prop' id='VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace'> <h6 id='proposition_7'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a> <a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivial</a> over <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subspace</a> of <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a> is <a class='existingWikiWord' href='/nlab/show/diff/pullback+bundle'>pullback</a> of bundle on <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_610' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a> and let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_611' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A \subset X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subspace</a>.</p> <p>If a topological vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_612' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>p</mi></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \overset{p}{\to} X</annotation></semantics></math> is such that its restriction <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_613' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>E\vert_A</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/trivial+vector+bundle'>trivializable</a>, then <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_614' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphic</a> to the <a class='existingWikiWord' href='/nlab/show/diff/pullback+bundle'>pullback bundle</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_615' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>q</mi> <mo>*</mo></msup><mi>E</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>q^\ast E'</annotation></semantics></math> of a topological vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_616' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>′</mo><mo>→</mo><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>E' \to X/A</annotation></semantics></math> over the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>.</p> </div> <div class='proof'> <h6 id='proof_15'>Proof</h6> <p>Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_617' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><munderover><mo>⟶</mo><mo>≃</mo><mi>ϕ</mi></munderover><mi>E</mi><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'> A \times k^n \underoverset{\simeq}{\phi}{\longrightarrow} E\vert_A </annotation></semantics></math></div> <p>be an isomorphism of vector bundles over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_618' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, which exists by assumption. Consider then on the total space <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_619' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>E\vert_A</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_620' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ϕ</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo><mo>∼</mo><msup><mi>ϕ</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>x</mi><mo>′</mo><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \phi^{-1}(x,v) \sim \phi^{-1}(x',v) </annotation></semantics></math></div> <p>for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_621' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>x</mi><mo>′</mo><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x,x' \in A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_622' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi><mo>∈</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>v \in k^n</annotation></semantics></math>. Let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_623' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>′</mo><mo>≔</mo><mi>E</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'> E' \coloneqq E/\sim </annotation></semantics></math></div> <p>be the corresponding <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a>. Observe that for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_624' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> we have <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_625' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><msub><mo>′</mo> <mi>x</mi></msub><mo>=</mo><msub><mi>E</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>E'_x = E_x</annotation></semantics></math> while for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_626' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x \in A</annotation></semantics></math> we have a canonical identification <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_627' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><msub><mo>′</mo> <mrow><mi>x</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow></msub><mo>≃</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>E'_{x/A} \simeq k^n</annotation></semantics></math>, and over these points quotient coprojection is identified with <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_628' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ϕ</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\phi^{-1}</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_629' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow /></mover></mtd> <mtd><mi>E</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>|</mo></mtd> <mtd><mi>x</mi><mo>∈</mo><mi>X</mi><mo>∖</mo><mi>A</mi></mtd></mtr> <mtr><mtd><msubsup><mi>ϕ</mi> <mi>x</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>|</mo></mtd> <mtd><mi>x</mi><mo>∈</mo><mi>A</mi></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ E &\overset{}{\longrightarrow}& E' \\ (x,v) &\mapsto& \left\{ \array{ (x,v) &\vert& x \in X \setminus A \\ \phi^{-1}_x(v) &\vert& x\in A } \right. } \,. </annotation></semantics></math></div> <p>Since the composite continuous function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_630' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mover><mo>⟶</mo><mi>p</mi></mover><mi>X</mi><mover><mo>⟶</mo><mi>q</mi></mover><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'> E \overset{p}{\longrightarrow} X \overset{q}{\longrightarrow} X/A </annotation></semantics></math></div> <p>respects the equivalence relation (in that it sends any two equivalent points to the same image point) the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the quotient space yields a continuous function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_631' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>′</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><mo>′</mo><mo>→</mo><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'> p' \;\colon\; E' \to X/A </annotation></semantics></math></div> <p>such that the following <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>diagram commutes</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_632' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi><mo>′</mo></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>p</mi></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><mi>p</mi><mo>′</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>q</mi></mover></mtd> <mtd><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ E &\longrightarrow& E' \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{p'}} \\ X &\overset{q}{\longrightarrow}& X/A } \,. </annotation></semantics></math></div> <p>We claim that this is a <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> diagram in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>:</p> <p>By the above description of the top horizontal function, it is a pullback diagram of underlying sets. Hence we need to see that the topology on <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_633' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base</a> given by the pre-images of the open subsets in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_634' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_635' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>E'</annotation></semantics></math>. Now by definition of the quotient space topology on <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_636' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>E'</annotation></semantics></math>, its open subsets are those of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_637' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> that either do not contain a point <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_638' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x,v)</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_639' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x \in A</annotation></semantics></math> or if they do, then they also contain all the points of the form <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_640' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>′</mo><mo>,</mo><msubsup><mi>ϕ</mi> <mrow><mi>x</mi><mo>′</mo></mrow> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><msub><mi>ϕ</mi> <mi>x</mi></msub><mo stretchy='false'>(</mo><mi>v</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x', \phi_{x'}^{-1}(\phi_x(v)))</annotation></semantics></math> for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_641' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>′</mo><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x' \in A</annotation></semantics></math>. Moreover, if <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_642' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x,v)</annotation></semantics></math> is in the open subset for <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_643' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x \in A</annotation></semantics></math>, then also <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_644' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>v</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x,v')</annotation></semantics></math> for all <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_645' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>v'</annotation></semantics></math> in some open ball in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_646' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>k^n</annotation></semantics></math> containing <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_647' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math>. Hence intersecing these pre-images with pre-images of open subsets of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_648' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> under <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_649' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> yields a basis for the topology.</p> <p>Hence it only remains to see that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_650' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>′</mo><mover><mo>⟶</mo><mrow><mi>p</mi><mo>′</mo></mrow></mover><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>E' \overset{p'}{\longrightarrow} X/A</annotation></semantics></math> is a vector bundle. The fiberwise linearity is clear, we need to show that it is locally trivializable.</p> <p>To that end, let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_651' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> be an open cover over which <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_652' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>p</mi></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \overset{p}{\to} X</annotation></semantics></math> has a local trivialization. Since <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_653' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A \subset X</annotation></semantics></math> is assumed to be closed, it follows that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_654' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∖</mo><mi>A</mi><mo>⊂</mo><mi>X</mi><mo>∖</mo><mi>A</mi><mo>}</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \left\{ U_i \setminus A \subset X \setminus A\right\}_{i \in I} </annotation></semantics></math></div> <p>is an open cover of the complement of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_655' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_656' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. By the nature of the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space topology</a>, this induces an open cover of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_657' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∖</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>X\setminus A</annotation></semantics></math>. If we adjoin the quotient <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_658' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>U/A</annotation></semantics></math> of an open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_659' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_660' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_661' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, then</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_662' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∖</mo><mi>A</mi><mo>⊂</mo><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi><mo stretchy='false'>}</mo><mo>⊔</mo><mo stretchy='false'>{</mo><mi>U</mi><mo stretchy='false'>/</mo><mi>A</mi><mo>⊂</mo><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> \{ U_i \setminus A \subset X/A \} \sqcup \{ U/A \subset X/A \} </annotation></semantics></math></div> <p>is an open cover of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_663' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>X/A</annotation></semantics></math>. Moreover, by the construction of <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_664' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>′</mo><mover><mo>→</mo><mrow><mi>p</mi><mo>′</mo></mrow></mover><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>E' \overset{p'}{\to} X/A</annotation></semantics></math> it is clear that this bundle has a local trivialization over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_665' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math>, since <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_666' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>p</mi></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \overset{p}{\to} X</annotation></semantics></math> does, and similarly <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_667' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>E'</annotation></semantics></math> trivializes over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_668' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>U/A</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_669' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> trivializes over <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_670' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>. But such a <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_671' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> does indeed exist by lemma <a class='maruku-ref' href='#IsomorphismOfVectorBundlesOnClosedSubsetOfCompactHausdorffSpaceExtendsToOpenNeighbourhoods'>4</a>.</p> </div> <div class='num_remark'> <h6 id='remark_7'>Remark</h6> <p>Prop <a class='maruku-ref' href='#VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace'>7</a> is the reason why <a class='existingWikiWord' href='/nlab/show/diff/reduced+K-theory'>reduced</a> <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a> satisfies the <a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+in+homology'>long exact sequences in cohomology</a> that make it a <a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'>generalized (Eilenberg-Steenrod) cohomology theory</a>. See</p> </div> <div class='num_prop' id='VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace'> <h6 id='proposition_8'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_672' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a> and <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_673' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A \subset X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> and write <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_674' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>X/A</annotation></semantics></math> for the corresponding <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> (<a href='quotient+space#QuotientBySubspace'>this example</a>) with quotient coprojection denoted <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_675' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>⟶</mo><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>q \colon X \longrightarrow X/A</annotation></semantics></math>.</p> <p>If <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_676' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible topological space</a> then the <a class='existingWikiWord' href='/nlab/show/diff/pullback+bundle'>pullback bundle</a> construction</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_677' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>q</mi> <mo>*</mo></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>Vect</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>/</mo><mi>A</mi><msub><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>/</mo><mo>∼</mo></mrow></msub><mo>⟶</mo><mi>Vect</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding='application/x-tex'> q^\ast \;\colon\; Vect(X/A)_{/\sim} \longrightarrow Vect(X)_{/\sim} </annotation></semantics></math></div> <p>is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>.</p> </div> <div class='proof'> <h6 id='proof_16'>Proof</h6> <p>By example <a class='maruku-ref' href='#TopologicalVectorBundleOverContractibleSpaceIsTrivializable'>16</a> every vector bundle <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_678' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>p</mi></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \overset{p}{\to} X</annotation></semantics></math> is trivializable over the contractible subspace <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_679' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. Therefore prop. <a class='maruku-ref' href='#VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace'>7</a> implies that it is in the image of the pullback bundle map <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_680' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>q</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>q^\ast</annotation></semantics></math>. This says that <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_681' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>q</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>q^\ast</annotation></semantics></math> is surjective. Finally, it is clear that it is injective. Therefore it is bijective.</p> </div> <div class='num_example'> <h6 id='example_17'>Example</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_682' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,x)</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed</a> <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact topological space</a>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_683' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>[0,1] \subset \mathbb{R}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a> with its <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>.</p> <p>There is</p> <ol> <li> <p>the ordinary <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, being the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_684' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X \times I</annotation></semantics></math></p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/reduced+cylinder'>reduced cylinder</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_685' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∧</mo><msub><mi>I</mi> <mo>+</mo></msub><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>×</mo><mi>I</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \wedge I_+ = (X \times I)/( \{x\} \times I )</annotation></semantics></math> which is the <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a> with the interval that has a base point freely adjoined</p> </li> </ol> <p>and</p> <ol> <li> <p>the ordinary <a class='existingWikiWord' href='/nlab/show/diff/suspension'>suspension</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_686' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mi>X</mi><mo>≔</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>S X \coloneqq (X \times I)/( X \times \{0,1\} )</annotation></semantics></math>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/reduced+suspension'>reduced suspension</a> <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_687' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi><mi>X</mi><mo>≔</mo><mo stretchy='false'>(</mo><mi>S</mi><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>×</mo><mi>I</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Sigma X \coloneqq (S X)/( \{x\} \times I )</annotation></semantics></math>.</p> </li> </ol> <p>In both cases the reduced space is obtained from the unreduced space by quotienting out the contractible closed subspace <math class='maruku-mathml' display='inline' id='mathml_f8cc9a97c4a020d82366513154bdcffc6b40b7a5_688' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>≃</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>I \simeq \{x\} \times I</annotation></semantics></math> and hence topological vector bundles do not see the difference between the reduced and the unreduced spaces, by prop. <a class='maruku-ref' href='#VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace'>8</a>.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+vector+bundle'>algebraic vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiable+vector+bundle'>differentiable vector bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> </ul> <h2 id='references'>References</h2> <p>The original reference for many results about bundles, including the theorem that <a class='existingWikiWord' href='/nlab/show/diff/concordance'>concordance</a> implies <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>, is</p> <ul> <li id='Steenrod'><a class='existingWikiWord' href='/nlab/show/diff/Norman+Steenrod'>Norman Steenrod</a>, <em>The Topology of Fibre Bundles</em>, Princeton University Press (1951, 1957, 1960) [[jstor:j.ctt1bpm9t5](https://www.jstor.org/stable/j.ctt1bpm9t5)]</li> </ul> <p>Further textbook accounts:</p> <ul> <li id='MilnorStasheff74'> <p><a class='existingWikiWord' href='/nlab/show/diff/John+Milnor'>John Milnor</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jim+Stasheff'>Jim Stasheff</a>, <em>Characteristic classes</em>, Princeton Univ. Press (1974) [[ISBN:9780691081229](https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76), <a href='https://doi.org/10.1515/9781400881826'>doi:10.1515/9781400881826</a>, <a href='https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf'>pdf</a>]</p> </li> <li> <p>Glenys Luke, Alexander S. Mishchenko, <em>Vector bundles and their applications</em>, Math. and its Appl. <strong>447</strong>, Kluwer 1998. viii+254 pp. <a href='http://www.ams.org/mathscinet-getitem?mr=99m:55019'>MR99m:55019</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dale+Husem%C3%B6ller'>Dale Husemoeller</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Joachim'>Michael Joachim</a>, <a class='existingWikiWord' href='/nlab/show/diff/Branislav+Jurco'>Branislav Jurco</a>, <a class='existingWikiWord' href='/nlab/show/diff/Martin+Schottenloher'>Martin Schottenloher</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Basic+Bundle+Theory+and+K-Cohomology+Invariants'>Basic Bundle Theory and K-Cohomology Invariants</a></em>, Lecture Notes in Physics, Springer 2008 (<a href='http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726corr1.pdf'>pdf</a>)</p> </li> </ul> <p>Lecture notes with an eye towards <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a>:</p> <ul> <li id='Wirthmuller12'> <p><a class='existingWikiWord' href='/nlab/show/diff/Klaus+Wirthm%C3%BCller'>Klaus Wirthmüller</a>, <em>Vector bundles and K-theory</em>, 2012 (<a class='existingWikiWord' href='/nlab/files/wirthmueller-vector-bundles-and-k-theory.pdf' title='pdf'>pdf</a>)</p> </li> <li id='Hatcher'> <p><a class='existingWikiWord' href='/nlab/show/diff/Allen+Hatcher'>Allen Hatcher</a>, chapter 1 of <em>Vector bundles and K-Theory</em>, (partly finished book) <a href='http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html'>web</a></p> </li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on January 13, 2025 at 20:03:20. See the <a href="/nlab/history/topological+vector+bundle" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/topological+vector+bundle" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/7791/#Item_20">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/topological+vector+bundle/47" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/topological+vector+bundle" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/topological+vector+bundle" accesskey="S" class="navlink" id="history" rel="nofollow">History (47 revisions)</a> <a href="/nlab/show/topological+vector+bundle/cite" style="color: black">Cite</a> <a href="/nlab/print/topological+vector+bundle" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/topological+vector+bundle" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>