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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 </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/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" 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topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/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/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/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/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/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/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/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/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/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/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/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/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/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/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="bundles">Bundles</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/bundles">bundles</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/parameterized+stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/slice+topos">slice topos</a>, <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/dependent+linear+type+theory">linear</a>) <a class="existingWikiWord" href="/nlab/show/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/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/retractive+space">retractive space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/numerable+bundle">numerable bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere+bundle">sphere bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+bundle">projective bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>, <a class="existingWikiWord" href="/nlab/show/stringor+bundle">stringor bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/2-gerbe">2-gerbe</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-gerbe">∞-gerbe</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+coefficient+bundle">local coefficient bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/2-vector+bundle">2-vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+vector+bundle">complex</a>/<a class="existingWikiWord" href="/nlab/show/holomorphic+vector+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/quaternionic+vector+bundle">quaternionic</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+vector+bundle">with connection</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphic</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+line+bundle">algebraic</a></p> <p><a class="existingWikiWord" href="/nlab/show/cubical+structure+on+a+line+bundle">cubical structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a><a class="existingWikiWord" href="/nlab/show/Vect%28X%29">of vector bundles</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/VectBund">VectBund</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum</a>, <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product</a>, <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product</a>, <a class="existingWikiWord" href="/nlab/show/inner+product+of+vector+bundles">inner product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+vector+bundle">dual vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+vector+bundle">stable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+of+spectra">bundle of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+bundle">natural bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/universal+principal+bundle">universal principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/universal+complex+line+bundle">universal complex line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a>, <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> </ul> <h2 id="sidebar_presentations">Presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microbundle">microbundle</a></p> </li> </ul> <h2 id="sidebar_examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+bundle">empty bundle</a>, <a class="existingWikiWord" href="/nlab/show/zero+bundle">zero bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological line bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/basic+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/Hopf+fibration">Hopf fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+line+bundle">canonical line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a>, <a class="existingWikiWord" href="/nlab/show/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/clutching+construction">clutching construction</a></li> </ul> </div></div> <h4 id="linear_algebra">Linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%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/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/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/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></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></li> <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><a href='#Examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <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><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/vector+bundle">vector bundle</a> in the context of <em><a class="existingWikiWord" href="/nlab/show/topology">topology</a></em>: a continuously varying collection of <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> over a given <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> <p>For more survey and motivation see at <em><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a></em>. Here we discuss the details of the general concept in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>. See also <em><a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable vector bundle</a></em> and <em><a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic vector bundle</a></em>.</p> <h2 id="definition">Definition</h2> <p>We first give the more abstract definiton in terms of <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a> (def. <a class="maruku-ref" href="#TopologicalVectorBundleInTermsOfSliceCategories"></a> below) and then unwind this to the traditional definition (def <a class="maruku-ref" href="#TopologicalVectorBundle"></a> below).</p> <p>In the following</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/topological+field">topological field</a></p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/real+numbers">real numbers</a></p> </li> <li> <p>or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/complex+numbers">complex numbers</a></p> </li> </ul> <p>equipped with the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>.</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a></em> means <em><a class="existingWikiWord" href="/nlab/show/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/Top">Top</a> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/slice+category">slice category</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/fiber+product">fiber product</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>, which we denote by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/field">field</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/structure">structure</a> of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/vector+space">vector space</a>-object <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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>which satisfy the <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a></p> </li> </ol> <p>such that</p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/local+trivialization">local triviality</a>) there exists</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/disjoint+union+space">disjoint union space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/isomorphism">isomorphism</a> of vector space objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><msub><mo>×</mo> <mi>I</mi></msub><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"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> U \times_I \mathbb{R}^{n} \overset{\simeq}{\longrightarrow} U \times_X E \,, </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \colon I \to \mathbb{N}</annotation></semantics></math> some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/indexed+set">indexed set</a> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>,</p> </li> </ol> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> </li> </ul> <p>It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is constant on <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Often this is required to be constant on all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and then called the <em><a class="existingWikiWord" href="/nlab/show/rank">rank</a></em> of the vector bundle.</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a></em> of topological vector bundles is simple a homomorphism of vector space objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/category">category</a>, usually denoted <a class="existingWikiWord" href="/nlab/show/Vect%28X%29">Vect(X)</a>.</p> </div> <p>Notice that viewed in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, the last condition means that there is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></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></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/pullback+square">pullback square</a> and the <a class="existingWikiWord" href="/nlab/show/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. Then a <em>topological vector bundle</em> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/finite-dimensional+vector+space">finite-dimensional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> on the <a class="existingWikiWord" href="/nlab/show/pre-image">pre-image</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/local+trivialization">locally trivial</a> in that there exists</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/homeomorphism">homeomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/product+topological+space">product topological space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/real+numbers">real numbers</a> (equipped with their <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>) to the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/linear+map">linear map</a> in each fiber in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/dimension">dimension</a>. In this case one says that the vector bundle has <em><a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>. (Over a <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected topological space</a> this is automatic, but the fiber dimension may be distinct over distinct <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a>.)</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/homomorphism">homomorphism</a></em> between them is</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> respects the <a class="existingWikiWord" href="/nlab/show/projections">projections</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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/Vect%28X%29">category of topological vector bundles</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, there is the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are the topological vector bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/morphisms">morphisms</a> are the topological vector bundle homomorphisms</p> </li> </ul> <p>according to def. <a class="maruku-ref" href="#TopologicalVectorBundle"></a>. This category usually denoted <a class="existingWikiWord" href="/nlab/show/Vect%28X%29">Vect(X)</a>.</p> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/set">set</a> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of this category.</p> </div> <p> <div class='num_remark' id='FiberwiseOperations'> <h6>Remark</h6> <p><strong>(fiberwise operations)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/FinDimVect">FinDimVect</a> of <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> over a <a class="existingWikiWord" href="/nlab/show/topological+field">topological</a> <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> is canonically a <a class="existingWikiWord" href="/nlab/show/Top">Top</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a>, and so are hence its <a class="existingWikiWord" href="/nlab/show/product+categories">product categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>FinDimVect</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">FinDimVect^{n}</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Top">Top</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/n-tuple">n-tuple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/fiber">fiber</a> over a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><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/dual+vector+spaces">dual vector spaces</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/dual+vector+bundle">dual vector bundle</a>;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mi>det</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><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/determinants">determinants</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/determinant+line+bundles">determinant line bundles</a>;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mo>⊕</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><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/direct+sum">direct sum</a> of vector space, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/direct+sum+of+vector+bundles">direct sum of vector bundles</a> (“<a class="existingWikiWord" href="/nlab/show/Whitney+sum">Whitney sum</a>”);</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mo>⊗</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><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/tensor+product+of+vector+spaces">tensor product of vector spaces</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/tensor+product+of+vector+bundles">tensor product of vector bundles</a>.</p> </li> </ul> <p></p> </div> </p> <div class="num_remark" id="TerminologyVectorBundles"> <h6 id="remark_2">Remark</h6> <p><strong>(some terminology)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>. Then:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/real+vector+bundles">real vector bundles</a></em>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/complex+vector+bundles">complex vector bundles</a></em>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/line+bundles">line bundles</a></em>, in particular of <em><a class="existingWikiWord" href="/nlab/show/real+line+bundles">real line bundles</a></em> and of <em><a class="existingWikiWord" href="/nlab/show/complex+line+bundles">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/local+trivialization">local trivialization</a> over a common <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>). Then there always exists an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/refinement">refinement</a></em> of these two covers is the open cover</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/product+topological+space">product topological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicalVectorBundle"></a>). This is called the <em><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></em> of <a class="existingWikiWord" href="/nlab/show/rank">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Given any topological vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/isomorphism">isomorphism</a> to a trivial bundle (if it exists)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/local+trivialization">local triviality</a> condition in the definition of topological vector bundles (def. <a class="maruku-ref" href="#TopologicalVectorBundle"></a>) says that they are locally isomorphic to the trivial vector bundle. One also says that the data consisting of an open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/homeomorphisms">homeomorphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></a> constitute a <em><a class="existingWikiWord" href="/nlab/show/local+trivialization">local trivialization</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/section">section</a> of a <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>).</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of vector bundles from the <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial</a> <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> (example <a class="maruku-ref" href="#TrivialTopologicalVectorBundle"></a>, remark <a class="maruku-ref" href="#TerminologyVectorBundles"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/sections">sections</a></em> (or <em>cross-sections</em>) of the vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>), then a <em>sub-bundle</em> is a homomorphism of topological vector bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><msub><mo stretchy="false">|</mo> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>E</mi><msub><mo>′</mo> <mi>x</mi></msub><mo>↪</mo><msub><mi>E</mi> <mi>x</mi></msub><mspace width="thinmathspace"></mspace><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/monomorphism">monomorphism</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/cocycles">cocycles</a> in <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a> constituted by their <a class="existingWikiWord" href="/nlab/show/transition+functions">transition functions</a>.</p> <div class="num_defn" id="ContinuousFunctionWithValuesInGLn"> <h6 id="definition_4">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> on <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> with values in the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/topological+group">topological group</a> with its standard <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a>, given as the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> via <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/compact-open+topology">compact-open topology</a> on the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a>. (That these topologies coincide is the statement of <a href="general+linear+group#AsSubspaceOfTheMappingSpace">this prop.</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/set">set</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/subspace+topology">subspace topology</a>), regarded as a <a class="existingWikiWord" href="/nlab/show/group">group</a> via the pointwise group operation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>g</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/sheaf">sheaf</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a>)</strong></p> <p>In the language of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> the assignment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> of continuous functions to open subsets and the restriction operations between these is called a <em><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of groups on the <a class="existingWikiWord" href="/nlab/show/site+of+open+subsets">site of open subsets</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/transition+functions">transition functions</a>)</strong></p> <p>Given a topological vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> and a choice of <a class="existingWikiWord" href="/nlab/show/local+trivialization">local trivialization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>) there are for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/continuous+functions">continuous functions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>g</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/general+linear+group">general linear group</a> (as in def. <a class="maruku-ref" href="#ContinuousFunctionWithValuesInGLn"></a>) given by composing the local trivialization isomorphisms:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <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"></mspace><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/transition+functions">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/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> <p>A <em>normalized <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cocycle</a> of degree 1 with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a></em> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>) is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a></p> </li> </ol> <p>such that</p> <ol> <li> <p>(normalization) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/constant+function">constant function</a> on the <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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><mspace width="thickmathspace"></mspace><mtext>on</mtext><mspace width="thinmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><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/disjoint+union">disjoint union</a> of all these cocycles as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/transition+functions">transition functions</a> are <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>).</p> <p>Then the set of induced <a class="existingWikiWord" href="/nlab/show/transition+functions">transition functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> is a <em>normalized Cech cocycle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><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> <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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><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> <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> <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"></mspace><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/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>, with open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/equivalence+relation">equivalence relation</a> on the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/disjoint+union+space">disjoint union space</a> of the patches <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/topological+subspaces">topological subspaces</a> with the <a class="existingWikiWord" href="/nlab/show/product+space">product space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>∼</mo><mspace width="thickmathspace"></mspace><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"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><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/quotient+topological+space">quotient topological space</a>. This comes with the evident projection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/continuous+function">continuous function</a> (by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a> construction, since the corresponding continuous <a class="existingWikiWord" href="/nlab/show/function">function</a> on the un-quotientd disjoint union space respects the equivalence relation). Moreover, each <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> of this map is identified with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></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/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> is a <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>)</strong></p> <p>Stated more <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretically</a>, the constructure of a topological vector bundle from Cech cocycle data in example <a class="maruku-ref" href="#TopologicalVectorBundleFromCechCocycle"></a> is a <a href="Top#UniversalConstructions">universal construction in topological spaces</a>, namely the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> of the two morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></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/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/transition+functions">transition functions</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/topological+vector+bundle">topological vector bundle</a> (def. <a class="maruku-ref" href="#TopologicalVectorBundle"></a>), let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/open+cover">open cover</a> of the base space, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/local+trivialization">local trivialization</a>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/transition+functions">transition functions</a> (def. <a class="maruku-ref" href="#TransitionFunctions"></a>). Then there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of vector bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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"></mspace><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"></a> to the original bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> (<a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in <a class="existingWikiWord" href="/nlab/show/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/equivalence+relation">equivalence relation</a> on the disjoint union space given by the transition functions, in that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> coprojection this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/extension">extends</a> to a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> on the quotient space such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/bijection">bijection</a>. Hence to show that it is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>, it is now sufficient to show that this is an <a class="existingWikiWord" href="/nlab/show/open+map">open map</a> (by <a href="Introduction+to+Topology+--+1#HomeoContinuousOpenBijection">this prop.</a>).</p> <p>So let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/quotient+topology">quotient topology</a> this means equivalently that its restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">O_i</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/subspace+topology">subspace topology</a>, this means that these images are open also in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. Therefore also the union <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/coboundary">coboundary</a> between <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> )</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> (def. <a class="maruku-ref" href="#CocycleCech"></a>), given by</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/open+covers">open covers</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>g</mi> <mo>′</mo></msub><msub><mo></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/coboundary">coboundary</a></em> between these two cocycles is</p> <ol> <li> <p>the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/functions">functions</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></mspace><mtext>and</mtext><mspace width="thinmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a></p> </li> </ol> <p>such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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><mspace width="thinmathspace"></mspace><mtext>on</mtext><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><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/diagrams">diagrams</a> of <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a> <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commute</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow> <mpadded width="0" lspace="-100%width"><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> <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"></mspace><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/refinement">refinement</a> of a <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> is a <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>, with respect to some open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/refinement">refinement</a> of the given open cover, hence an open cover such that there exists a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">c'</annotation></semantics></math> which is cohomologous to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/isomorphism">isomorphism</a> of topological vector bundles induces <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a> between their <a class="existingWikiWord" href="/nlab/show/transition+functions">transition functions</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a topological space, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>.</p> <p>Every <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of topological vector bundles</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></a> induces a coboundary between the two cocycles,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>∼</mo><msub><mi>c</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><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"></a>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By example <a class="maruku-ref" href="#FinerCoverCech"></a> we may assume without restriction that the two Cech cocycles are defined with respect to the same open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> the two bundles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/commuting+diagram">diagram commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">c_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow> <mpadded width="0" lspace="-100%width"><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> <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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></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/Cech+cohomology">Cech cohomology</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. The relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∼</mo></mrow><annotation encoding="application/x-tex">\sim</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cocycles</a> of being cohomologous (def. <a class="maruku-ref" href="#CoboundaryCech"></a>) is an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Cech+cohomology">Cech</a> <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> (def. <a class="maruku-ref" href="#CocycleCech"></a>).</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><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/equivalence+classes">equivalence classes</a>. This is called the <em><a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Cech+cohomology">Cech cohomology</a> computes <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/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"></a>) constitutes a <a class="existingWikiWord" href="/nlab/show/natural+bijection">bijection</a> between the degree-1 Cech cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>) and the set of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of topological vector bundles on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicalVectorBundle"></a>, remark <a class="maruku-ref" href="#TopologicalVectorBundlesCategory"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>), then the vector bundles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>κ</mi> <mi>α</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow> <mpadded width="0" lspace="-100%width"><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> <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></mrow> <mpadded width="0" lspace="-100%width"><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> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∃</mo><mo>!</mo></mrow></mpadded></msup></mtd> <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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> the function is <a class="existingWikiWord" href="/nlab/show/surjective+function">surjective</a>, and by prop. <a class="maruku-ref" href="#CechCoboundaryFromIsomorphismBetweenVectoreBundles"></a> it is injective.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></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/tautological+line+bundle">tautological line bundle</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/projective+space">projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/tautological+line+bundle">tautological line bundle</a></em> whose fiber over the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/cylinder">cylinder</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><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"></mspace><mo>⊂</mo><mspace width="thinmathspace"></mspace><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/circle">circle</a> with its <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial</a> <a class="existingWikiWord" href="/nlab/show/real+vector+bundle">real</a> <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> on the circle is the the <a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/Moebius+strip">Moebius strip</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><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"></mspace><mo>⊂</mo><mspace width="thinmathspace"></mspace><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/circle">circle</a> with its <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>. Consider the <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Cech+cohomology">Cech cohomology</a> cocycle (remark <a class="maruku-ref" href="#CechCoycleCondition"></a>) on this cover by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></a> these functions define a vector bundle. This is the <em><a class="existingWikiWord" href="/nlab/show/Moebius+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/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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><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/2-sphere">2-sphere</a> with its <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>. Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/complements">complements</a> of antipodal points</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/complex+line+bundle">complex line bundle</a> this defined is called the <em><a class="existingWikiWord" href="/nlab/show/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/complex+projective+space">complex projective space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Riemann+sphere">Riemann sphere</a>), the basic complex line bundle is the <a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological line bundle</a> (example <a class="maruku-ref" href="#TautologicalLineBundle"></a>) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/clutching+construction">clutching construction</a>)</strong></p> <p>Generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/n-sphere">n-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/hemispheres">hemispheres</a> intersecting in an <a class="existingWikiWord" href="/nlab/show/equator">equator</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>−</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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/clutching+construction">clutching construction</a></em> of vector bundles over <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>.</p> </div> <div class="num_example"> <h6 id="example_12">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> underlyithe ng a <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a> then its <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/real+vector+bundle">real vector bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank</a> is the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/normal+bundle">normal bundle</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/embedding+of+differentiable+manifolds">embedding of differentiable manifolds</a>, then the <em><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/quotient+vector+space">quotient vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>).</p> <p>If a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of vector bundles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/fiber">fiber</a> over each point to a linear isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo stretchy="false">|</mo> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has an <a class="existingWikiWord" href="/nlab/show/inverse+function">inverse function</a> of underlying sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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></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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: Over each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> we find an open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/trivial+vector+bundles">trivial vector bundles</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></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> <mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></msup><mo stretchy="false">↓</mo></mtd> <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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/homeomorphisms">homeomorphisms</a>, hence that their inverse functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> and since they fix the base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/locally+compact+topological+space">locally compact</a> (like every <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional vector space</a>, by the <a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a>), the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <a class="existingWikiWord" href="/nlab/show/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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>h</mi><mo stretchy="false">˜</mo></mover> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/compact-open+topology">compact-open topology</a>) which factors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/evaluation">evaluation</a> map as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/general+linear+group">general linear group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/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/inverse+matrices">inverse matrices</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/rational+function">rational function</a> on its domain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Euclidean+space">Euclidean space</a> and since <a class="existingWikiWord" href="/nlab/show/rational+functions+are+continuous">rational functions are continuous</a> on their domain of definition, it follows that the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> to find an isomorphism between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/quotient+topological+space">quotient topological space</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> (the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>) the set of continuous functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mtd> <mtd><msub><mi>E</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><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"></mspace><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/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/extension">extend</a> to a unique continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/commuting+diagram">diagram commutes</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/rank+of+a+vector+bundle">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> admits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sections">sections</a> (example <a class="maruku-ref" href="#VectorBundleSections"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>σ</mi> <mi>k</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>). In fact, with the sections regarded as vector bundle homomorphisms out of the trivial vector bundle of rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>), these sections <em>are</em> the trivialization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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"></a>.</p> </div> <h3 id="DirectSummandBundles">Direct summand bundles</h3> <p>We discuss properties of the <a class="existingWikiWord" href="/nlab/show/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/paracompact+spaces">paracompact spaces</a> are <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct summands</a>)</strong></p> <p>Let</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+space">paracompact Hausdorff space</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/topological+vector+bundle">topological vector bundle</a> (def. <a class="maruku-ref" href="#TopologicalVectorBundle"></a>).</p> </li> </ol> <p>Then every topological vector sub-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>) is a direct vector bundle summand, in that there exists another vector sub-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>) such that their <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/inner+product+on+vector+bundles">inner product on vector bundles</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/orthogonal+complement">orthogonal complement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/subspace">subspace</a> of these orthogonal complements is readily checked to be a <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><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"></mspace><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/compact+Hausdorff+space">compact Hausdorff space</a> are <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct summands</a> of a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>)</strong></p> <p>Let</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/topological+vector+bundle">topological vector bundle</a> (def. <a class="maruku-ref" href="#TopologicalVectorBundle"></a>).</p> </li> </ol> <p>Then there exists another topological vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/direct+sum+of+vector+bundles">direct sum of vector bundles</a> of the two is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>⊕</mo><mover><mi>E</mi><mo stretchy="false">˜</mo></mover><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/open+cover">open cover</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> over which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/local+trivialization">local trivialization</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/finite+cover">finite sub-cover</a>, hence a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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/partition+of+unity">partition of unity</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>f</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/support">support</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>E</mi><mo>⟶</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><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/direct+sum">direct sum</a> of these yields a vector bundle homomorphism of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>.</p> </div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>Prop. <a class="maruku-ref" href="#TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle"></a> is key in the analysis of <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> groups on <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">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/concordance">concordance</a> of topological vector bundles over a <a class="existingWikiWord" href="/nlab/show/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"></a> below). In particular this implies tht the <a class="existingWikiWord" href="/nlab/show/pullbacks+of+vector+bundles">pullbacks of vector bundles</a> along two <a class="existingWikiWord" href="/nlab/show/homotopy">homotopic</a> <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> are <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> (corollary <a class="maruku-ref" href="#PullbackOfvectorBundlesAlongHomotopicMapsAreIsomorphic"></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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/product+topological+space">product topological space</a> with the <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/projections">projections</a> out of the product space.</p> <div class="num_lemma" id="TrivilizationOfVectorBundleOverProductSpaceWithInterval"> <h6 id="lemma_2">Lemma</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, then a vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>) if its restrictions to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, then for every topological vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> such that the vector bundle trivializes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/local+trivialization">local trvializability</a> of the vector bundle, there exists an open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/compact+topological+space">compact topological space</a> (for instance by the <a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a>) and hence there exists a <a class="existingWikiWord" href="/nlab/show/finite+set">finite</a> <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/intersection">intersection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Moreover</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/metric+topology">metric topology</a> each <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/refinement">refinement</a> of this cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/interval">interval</a>. Since this is a finite cover, we may find numbers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> this implies that the vector bundle in fact trivializes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/concordance">concordance</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+space">paracompact Hausdorff space</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/topological+vector+bundle">topological vector bundle</a> over the <a class="existingWikiWord" href="/nlab/show/product+space">product space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a> (hence a <em><a class="existingWikiWord" href="/nlab/show/concordance">concordance</a></em> of topological vector bundles on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/isomorphism">isomorphic</a> vector bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> there exists an open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> such that the vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> trivializes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/countable+cover">countable cover</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/disjoint+union">disjoint union</a> of open subsets that each are contained in one of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> still trivializes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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/partition+of+unity">partition of unity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/graph">graph</a> of the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/subspace+topology">subspace topology</a>, and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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> <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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow></mover></mtd> <mtd><msub><mi>X</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd></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/continuous+functions">continuous functions</a>: By the nature of the <a class="existingWikiWord" href="/nlab/show/product+topology">product topology</a> and the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> it is sufficient to check for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: To see local trivializability over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> choose a local trivialization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> over some open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/fiber+product">fiber product</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/pasting+law">pasting law</a> says that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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> <mtd><mover><mo>⟶</mo><mrow><msub><mi>h</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>E</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <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> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mi>n</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a> implies that each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <p>By local finiteness, each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+space">paracompact Hausdorff space</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/vector+bundle">vector bundle</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/continuous+functions">continuous functions</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/left+homotopy">left homotopy</a> between them. Then there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of vector bundles over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> between the <a class="existingWikiWord" href="/nlab/show/pullback+of+vector+bundles">pullback of vector bundles</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>, respectively:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/left+homotopy">left homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; X \times [0,1] \longrightarrow Y \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/pasting+law">pasting law</a> for pullbacks we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></a>.</p> </div> <div class="num_example" id="HomotopyInvarianceOfIsomorphismClassesOfVectorBundles"> <h6 id="example_15">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopy+invariance">homotopy invariance</a> of isomorphism classes of vector bundles)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces">paracompact Hausdorff spaces</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/continuous+function">continuous function</a> which is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>. Then pullback along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> constitutes a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> on sets of isomorphism classes of topological vector bundles:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/left+homotopies">left homotopies</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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"></a> implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/inverse+function">inverse function</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/contractible+topological+space">contractible topological space</a>, then every topological vector bundle over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/trivial+vector+bundle">trivial vector bundle</a>.</p> </div> <div class="proof"> <h6 id="proof_13">Proof</h6> <p>That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/left+homotopy">left homotopy</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mn>0</mn></msub></mrow></mpadded><mo stretchy="false">↓</mo></mtd> <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 width="0" lspace="-100%width"><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></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><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"></a> it follows that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/pullback+bundle">pullback bundle</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/trivial+vector+bundles">trivial vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/closed+subspaces">closed subspaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/compact+Hausdorff+spaces">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/isomorphism">isomorphism</a> of vector bundles on <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a> of <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a> <a class="existingWikiWord" href="/nlab/show/extension">extends</a> to <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>) of the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product of vector bundles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/dual+vector+bundle">dual vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/open+cover">open cover</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/compact+Hausdorff+spaces+are+normal">compact Hausdorff spaces are normal</a>, the <a class="existingWikiWord" href="/nlab/show/shrinking+lemma">shrinking lemma</a> applies and gives a refinement of this by a cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/matrices">matrices</a>, this is a set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/compact+Hausdorff+spaces+are+normal">compact Hausdorff spaces are normal</a> the <a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a> therefore applies to these component functions and yields <a class="existingWikiWord" href="/nlab/show/extensions">extensions</a> of each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/continuous+function">continuous function</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>σ</mi><mo stretchy="false">^</mo></mover> <mi>i</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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"></mspace><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/paracompact+Hausdorff+spaces">paracompact Hausdorff spaces</a>, and since <a class="existingWikiWord" href="/nlab/show/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/partition+of+unity">partition of unity</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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/extension">extension</a> of the original section, in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \hat \sigma\vert_A = \sigma \,. </annotation></semantics></math></div> <p>This is because for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><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> <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> <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> <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> <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> <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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/open+subset">open subset</a> of the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>, hence such that there is an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/vector+bundle">vector bundle</a> <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial</a> over <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a> of <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> is <a class="existingWikiWord" href="/nlab/show/pullback+bundle">pullback</a> of bundle on <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/closed+subspace">closed subspace</a>.</p> <p>If a topological vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/trivializable+vector+bundle">trivializable</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/pullback+bundle">pullback bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/quotient+space">quotient space</a>.</p> </div> <div class="proof"> <h6 id="proof_15">Proof</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/equivalence+relation">equivalence relation</a> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/quotient+topological+space">quotient topological space</a>. Observe that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow></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"></mspace><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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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/universal+property">universal property</a> of the quotient space yields a continuous function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>′</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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></mrow> <mpadded width="0" lspace="-100%width"><mi>p</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <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"></mspace><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/pullback">pullback</a> diagram in <a class="existingWikiWord" href="/nlab/show/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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/base+for+a+topology">base</a> given by the pre-images of the open subsets in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> that either do not contain a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">v'</annotation></semantics></math> in some open ball in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">k^n</annotation></semantics></math> containing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/quotient+space+topology">quotient space topology</a>, this induces an open cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math>, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">E'</annotation></semantics></math> trivializes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> trivializes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. But such a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>.</p> </div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>Prop <a class="maruku-ref" href="#VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace"></a> is the reason why <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced</a> <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> satisfies the <a class="existingWikiWord" href="/nlab/show/long+exact+sequences+in+cohomology">long exact sequences in cohomology</a> that make it a <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology+theory">generalized (Eilenberg-Steenrod) cohomology theory</a>. See</p> </div> <div class="num_prop" id="VectorBundlesOverQuotientByContractibleSubspaceAreEquivalentToVectorBundlesOnTotalSpace"> <h6 id="proposition_8">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/quotient+topological+space">quotient topological space</a> (<a href="quotient+space#QuotientBySubspace">this example</a>) with quotient coprojection denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/contractible+topological+space">contractible topological space</a> then the <a class="existingWikiWord" href="/nlab/show/pullback+bundle">pullback bundle</a> construction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>q</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><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/isomorphism">isomorphism</a>.</p> </div> <div class="proof"> <h6 id="proof_16">Proof</h6> <p>By example <a class="maruku-ref" href="#TopologicalVectorBundleOverContractibleSpaceIsTrivializable"></a> every vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. Therefore prop. <a class="maruku-ref" href="#VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace"></a> implies that it is in the image of the pullback bundle map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/closed+interval">closed interval</a> with its <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>.</p> <p>There is</p> <ol> <li> <p>the ordinary <a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, being the <a class="existingWikiWord" href="/nlab/show/product+space">product space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/reduced+cylinder">reduced cylinder</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/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/suspension">suspension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/reduced+suspension">reduced suspension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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"></a>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+vector+bundle">algebraic vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+vector+bundle">differentiable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/concordance">concordance</a> implies <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, is</p> <ul> <li id="Steenrod"><a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>, <em>The Topology of Fibre Bundles</em>, Princeton University Press (1951, 1957, 1960) [<a href="https://www.jstor.org/stable/j.ctt1bpm9t5">jstor:j.ctt1bpm9t5</a>]</li> </ul> <p>Further textbook accounts:</p> <ul> <li id="MilnorStasheff74"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Characteristic classes</em>, Princeton Univ. Press (1974) [<a href="https://press.princeton.edu/books/paperback/9780691081229/characteristic-classes-am-76-volume-76">ISBN:9780691081229</a>, <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/Dale+Husemoeller">Dale Husemoeller</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Joachim">Michael Joachim</a>, <a class="existingWikiWord" href="/nlab/show/Branislav+Jurco">Branislav Jurco</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Schottenloher">Martin Schottenloher</a>, <em><a class="existingWikiWord" href="/nlab/show/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/topological+K-theory">topological K-theory</a>:</p> <ul> <li id="Wirthmuller12"> <p><a class="existingWikiWord" href="/nlab/show/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/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> </body></html> </div> <div class="revisedby"> <p> Last revised on January 13, 2025 at 20:03:20. 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