CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> proper homotopy theory in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> proper homotopy theory </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/discussions/?CategoryID=0" title="Discuss this page on the nForum. It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homotopy_theory">Homotopy theory</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> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential 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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea_of_some_problems'>Idea of some problems</a></li> <li><a href='#intuitions'>Intuitions</a></li> <ul> <li><a href='#ends'>Ends</a></li> <ul> <li><a href='#examples'>Examples</a></li> </ul> <li><a href='#proper_maps'>Proper maps</a></li> <li><a href='#end_spaces_continued'>End spaces continued</a></li> <li><a href='#proper_homotopy'>Proper homotopy</a></li> <li><a href='#germs_at_'>Germs at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></a></li> </ul> <li><a href='#proper_analogues_of_the_fundamental_group'>Proper analogues of the fundamental group</a></li> <ul> <li><a href='#first_attempt'>First attempt</a></li> <li><a href='#waldhausen_boundary'>Waldhausen boundary</a></li> <li><a href='#second_attempt'>Second attempt</a></li> <li><a href='#browngrossman_homotopy_groups'>Brown–Grossman homotopy groups</a></li> </ul> <li><a href='#to_do'>To do!</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea_of_some_problems">Idea of some problems</h2> <p><strong>Proper homotopy theory</strong> is both an old and a fairly new area of <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>. It deals with properties of non-<a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s, that cannot be detected using maps from <a class="existingWikiWord" href="/nlab/show/simplex">simplices</a> in these spaces, such as used in the <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoid">singular complex</a>.</p> <p>Historically the subject is traced back to Kerekjarto’s classification of non-compact surfaces in 1923, but its emergence as an important tool in geometric topology came with <a class="existingWikiWord" href="/nlab/show/Larry+Siebenmann">Larry Siebenmann</a>‘s work in 1965.</p> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a <em>smooth</em> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> with boundary, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">\partial M</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>∖</mo><mo>∂</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">M\setminus \partial M</annotation></semantics></math> is an open manifold. Now suppose someone gives us an open manifold <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 it possible to detect if there is a compact manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>∖</mo><mo>∂</mo><mi>M</mi><mo>≅</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">M\setminus \partial M \cong N</annotation></semantics></math>. Siebenmann showed that certain conditions on the <em>ends</em> of <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> were necessary and that there were obstructions if the dimension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> was greater than 5.</p> <p>Its potential importance of proper homotopy, for example for physical applications, comes from the fact that the phenomena it studies include the limiting behaviour of the system.</p> <h2 id="intuitions">Intuitions</h2> <p>The basic hypothesis will be 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> will be a <a class="existingWikiWord" href="/nlab/show/connected+space">connected and locally connected</a> compact <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>. It will usually be <a class="existingWikiWord" href="/nlab/show/sigma-compact">sigma-compact</a>, i.e., there will be an increasing sequence, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>K</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{K_n\}</annotation></semantics></math>, of compact subspaces with each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">K_n</annotation></semantics></math> in the interior of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">K_{n+1}</annotation></semantics></math> and such that</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><munderover><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo> <mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow> <mn>∞</mn></munderover><msub><mi>K</mi> <mi>n</mi></msub><mo>,</mo></mrow><annotation encoding="application/x-tex">X = \bigcup^\infty_{n=0}K_n,</annotation></semantics></math></div> <p>These spaces will most often than not be locally finite <a class="existingWikiWord" href="/nlab/show/simplicial+complex">simplicial complex</a>es.</p> <p>We will be interested in the <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> of such spaces ‘out towards its ends’</p> <h3 id="ends">Ends</h3> <p>To illustrate the idea of the <a class="existingWikiWord" href="/nlab/show/end+of+a+topological+space">ends</a> of a space <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>, we note that naively <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> has two ends, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">-\infty</annotation></semantics></math>, whilst <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> has only one as it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>∖</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">S^2\setminus \{\infty\}</annotation></semantics></math>, (but that is vague!).</p> <p>More exactly, consider the system of spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ε</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>closure</mi><mo stretchy="false">(</mo><mi>X</mi><mo>∖</mo><mi>K</mi><mo stretchy="false">)</mo><mo>:</mo><mi>K</mi><mi>compact</mi><mo>⊂</mo><mi>X</mi><mo stretchy="false">}</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">\varepsilon(X) = \{closure(X\setminus K) : K compact \subset X\},</annotation></semantics></math></div> <p>This is an inverse system or <a class="existingWikiWord" href="/nlab/show/pro-object">pro-object</a> in the category of spaces. Applying the connected component functor, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math>, to this system of spaces gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(X)</annotation></semantics></math>, and, classically, one takes the limit of this to get</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>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>lim</mi><msub><mi>π</mi> <mn>0</mn></msub><mi>ε</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">e(X) = lim \pi_0\varepsilon(X),</annotation></semantics></math></div> <p>the <em>set of ends</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>. In general, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(X)</annotation></semantics></math> would be given the <span class="newWikiWord">inverse limit topology<a href="/nlab/new/inverse+limit+topology">?</a></span>, to preserve more of the information coming from its construction. This space is the <em>space of (Freudenthal) ends 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></em>. It is a <a class="existingWikiWord" href="/nlab/show/profinite+space">profinite space</a>.</p> <h4 id="examples">Examples</h4> <ul> <li> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">X_8</annotation></semantics></math> be the figure eight space, the one-point union of two circles, and 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 its universal cover. This is an infinite ‘thorn bush’. It has infinitely many ends and</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>X</mi><mo stretchy="false">)</mo><mo>≅</mo><msup><mn>2</mn> <mrow><msub><mi>ℵ</mi> <mn>0</mn></msub></mrow></msup><mo>.</mo></mrow><annotation encoding="application/x-tex">e(X) \cong 2^{\aleph_0}.</annotation></semantics></math></div></li> <li> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> be a compact manifold and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>M</mi><mo>∖</mo><mo>∂</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X = M\setminus \partial M</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≅</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mo>∂</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(X) \cong \pi_0(\partial M)</annotation></semantics></math>.</p> </li> </ul> <h3 id="proper_maps">Proper maps</h3> <p>The assignment sending <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(X)</annotation></semantics></math> cannot be functorial on the category of spaces and <em>continuous</em> maps, since the contracting map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>→</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}\to \{0\}</annotation></semantics></math> is continuous, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(\mathbb{R})</annotation></semantics></math> is <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><mn>∞</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-\infty, \infty\}</annotation></semantics></math>, whilst <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(\{0\})</annotation></semantics></math> is empty since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0\}</annotation></semantics></math> is compact. The problem is that continuity is really about inverse images (inverse image of open is open), but the inverse image mapping does not preserve compactness (as in the example!).</p> <div class="un_defn"> <h6 id="definition_recall">Definition (recall)</h6> <p>A map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:X\to Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a> if for each subset <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> compact in <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> then <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 stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(K)</annotation></semantics></math> is compact 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>.</p> </div> <h3 id="end_spaces_continued">End spaces continued</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:X\to Y</annotation></semantics></math> is proper, then it induces a pro-morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ε</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>ε</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ε</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\varepsilon(f) : \varepsilon(X)\to \varepsilon(Y)</annotation></semantics></math></div> <p>and hence a <em>continuous map of the end spaces</em></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>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>e</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>,</mo></mrow><annotation encoding="application/x-tex">e(f):e(X)\to e(Y),</annotation></semantics></math></div> <p>and <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> becomes a functor from some category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Proper</mi></mrow><annotation encoding="application/x-tex">Proper</annotation></semantics></math> of spaces and proper maps to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Stone</mi></mrow><annotation encoding="application/x-tex">Stone</annotation></semantics></math>, the category of <a class="existingWikiWord" href="/nlab/show/profinite+space">profinite space</a>s / <a class="existingWikiWord" href="/nlab/show/Stone+space">Stone space</a>s.</p> <h3 id="proper_homotopy">Proper homotopy</h3> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>For any space <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 natural inclusions 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> into <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_i(x) = (x,i)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i = 0,1</annotation></semantics></math> together with the projection maps from <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> 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>, are proper maps.</p> </div> <div class="un_cor"> <h6 id="corollary">Corollary</h6> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Proper</mi></mrow><annotation encoding="application/x-tex">Proper</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/cylinder+functor">cylinder functor</a>.</p> </div> <p>We call the corresponding notion of homotopy, ‘proper homotopy’. We get a ‘Proper category’ and an associated ‘proper homotopy category’, which we will denote <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Proper</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Proper)</annotation></semantics></math>. If we are restricting to <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>-compact spaces we may write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Proper</mi> <mi>σ</mi></msub></mrow><annotation encoding="application/x-tex">Proper_\sigma</annotation></semantics></math>, and so on.</p> <h3 id="germs_at_">Germs at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></h3> <p>Although a proper map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f: X\to Y</annotation></semantics></math> will induce a continuous <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ε</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\varepsilon(f)</annotation></semantics></math>, on the end spaces, 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> does not need to be defined on the whole 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> for this to work, as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ε</mi></mrow><annotation encoding="application/x-tex">\varepsilon</annotation></semantics></math> encodes behaviour ‘out towards <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>’. This leads to the notion of a ‘germ at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>’.</p> <p>Suppose <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 locally compact Hausdorff 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>. The inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">j: A\to X</annotation></semantics></math> is ‘cofinal’ if the closure 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> is compact. Note that a cofinal inclusion is proper and induces an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ε</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≅</mo><mi>ε</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\varepsilon(A)\cong \varepsilon(X)</annotation></semantics></math>. Let <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> be the class of all cofinal inclusions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Proper</mi></mrow><annotation encoding="application/x-tex">Proper</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Proper</mi> <mn>∞</mn></msub><mo>=</mo><mi>Proper</mi><mo stretchy="false">[</mo><msup><mi>Σ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Proper_\infty = Proper[\Sigma^{-1}]</annotation></semantics></math>, the category obtained by formally inverting the cofinal inclusions.</p> <div class="un_defn"> <h6 id="definition">Definition</h6> <p>This category is called the <em>proper category at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></em>.</p> </div> <p>Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Proper</mi><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Proper,\Sigma)</annotation></semantics></math> admits a <a class="existingWikiWord" href="/nlab/show/calculus+of+fractions">calculus of right fractions</a>, so any morphism from <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 <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Proper</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">Proper_\infty</annotation></semantics></math> can be represented by a diagram</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><mi>j</mi></mover><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">X\stackrel{j}{\leftarrow} A \stackrel{f}{\rightarrow}Y,</annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> a cofinal inclusion, i.e., <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 defined on some ‘neighbourhood of the end 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> <p>Two such diagrams</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><msup><mi>j</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow></mover><msup><mi>A</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mover><mo>→</mo><mrow><msup><mi>f</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow></mover><mi>Y</mi><mi>and</mi><mi>X</mi><mover><mo>←</mo><mrow><msup><mi>j</mi> <mrow><mstyle scriptlevel="0"><mo>′</mo></mstyle><mstyle scriptlevel="0"><mo>′</mo></mstyle></mrow></msup></mrow></mover><msup><mi>A</mi> <mrow><mstyle scriptlevel="0"><mo>′</mo></mstyle><mstyle scriptlevel="0"><mo>′</mo></mstyle></mrow></msup><mover><mo>→</mo><mrow><msup><mi>f</mi> <mrow><mstyle scriptlevel="0"><mo>′</mo></mstyle><mstyle scriptlevel="0"><mo>′</mo></mstyle></mrow></msup></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\stackrel{j^{\prime}}{\leftarrow} A^{\prime} \stackrel{f^\prime}{\rightarrow}Y and X\stackrel{j^{\prime\prime}}{\leftarrow} A^{\prime\prime} \stackrel{f^{\prime\prime}}{\rightarrow}Y</annotation></semantics></math></div> <p>represent the same germ if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo stretchy="false">|</mo><mi>A</mi><mo>=</mo><msup><mi>f</mi> <mrow><mstyle scriptlevel="0"><mo>′</mo></mstyle><mstyle scriptlevel="0"><mo>′</mo></mstyle></mrow></msup><mo stretchy="false">|</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f^\prime | A = f^{\prime\prime}|A</annotation></semantics></math> for some cofinal 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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo>∪</mo><msup><mi>A</mi> <mrow><mstyle scriptlevel="0"><mo>′</mo></mstyle><mstyle scriptlevel="0"><mo>′</mo></mstyle></mrow></msup><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A^\prime \cup A^{\prime\prime}\subset A</annotation></semantics></math>.</p> <p>There is also a homotopy category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Proper</mi> <mn>∞</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Proper_\infty)</annotation></semantics></math></p> <div class="query"> <p>The end space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(X)</annotation></semantics></math> is a Stone space so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Max</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Max(R)</annotation></semantics></math> for some Boolean algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> (Stone duality) In the 1960s someone (Goldman?) looked at a ring, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, of ‘almost continuous functions’ from <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>, that gave the right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(X)</annotation></semantics></math>. Can this idea help integrate better the ideas of proper homotopy etc. with modern methods of algebraic geometry?</p> </div> <h2 id="proper_analogues_of_the_fundamental_group">Proper analogues of the fundamental group</h2> <p>The end space behaves a bit like a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math> and usually spaces will have many ends, so are not ‘connected at infinity’. If we try to do a fundamental group or groupoid analogue, this means life will get more complicated. We will try with the assumption of a space having a single end for simplicity! We will also assume <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 <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>-compact.</p> <h3 id="first_attempt">First attempt</h3> <p>We could try defining <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(\varepsilon(X))</annotation></semantics></math> as a progroup, then taking its limit. For this we would take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>K</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{K_n\}</annotation></semantics></math> an exhausting increasing sequence of compact subsets and setting <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><mo>∖</mo><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i = X\setminus K_i</annotation></semantics></math>, pick a base point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math> in 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></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math>, and we will get groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(U_i,x_i)</annotation></semantics></math>. We however need induced homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>x</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(U_{i+1},x_{i+1})\to \pi_1(U_i,x_i)</annotation></semantics></math>, and for this we have to choose an arc in <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> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">x_{i+1}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math>. We can combine these to get a <em>base ray</em>, rather than a <em>base point</em>, that is, we need a proper map, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\alpha : [0,\infty)\to X</annotation></semantics></math>. With that we do get an inverse sequence of groups, but there are problems. What is the dependence of the inverse system on the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>?</p> <svg xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:cc="http://creativecommons.org/ns#" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns="http://www.w3.org/2000/svg" xmlns:sodipodi="http://sodipodi.sourceforge.net/DTD/sodipodi-0.dtd" xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape" width="560.23041" height="312.2442" id="svg2406" sodipodi:version="0.32" inkscape:version="0.46" sodipodi:docname="cylinder.svg" inkscape:output_extension="org.inkscape.output.svg.inkscape" version="1.0"> <defs id="defs2408"> <inkscape:perspective sodipodi:type="inkscape:persp3d" inkscape:vp_x="0 : 526.18109 : 1" inkscape:vp_y="0 : 1000 : 0" inkscape:vp_z="744.09448 : 526.18109 : 1" inkscape:persp3d-origin="372.04724 : 350.78739 : 1" id="perspective2414"></inkscape:perspective> </defs> <sodipodi:namedview id="base" pagecolor="#ffffff" bordercolor="#666666" borderopacity="1.0" gridtolerance="10000" guidetolerance="10" objecttolerance="10" inkscape:pageopacity="0.0" inkscape:pageshadow="2" inkscape:zoom="1.55625" inkscape:cx="762.19162" inkscape:cy="154.01741" inkscape:document-units="px" inkscape:current-layer="layer1" showgrid="false" inkscape:window-width="1152" inkscape:window-height="701" inkscape:window-x="0" inkscape:window-y="22"></sodipodi:namedview> <metadata id="metadata2411"> <rdf:RDF> <cc:Work rdf:about=""> <dc:format>image/svg+xml</dc:format> <dc:type rdf:resource="http://purl.org/dc/dcmitype/StillImage"></dc:type> </cc:Work> </rdf:RDF> </metadata> <g inkscape:label="Layer 1" inkscape:groupmode="layer" id="layer1" transform="translate(-79.884809,-93.576859)"> <path sodipodi:type="arc" style="opacity:1;fill:#cccccc;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate" id="path2383" sodipodi:cx="168.57143" sodipodi:cy="254.09448" sodipodi:rx="34.285713" sodipodi:ry="61.42857" d="M 202.85714,254.09448 A 34.285713,61.42857 0 1 1 134.28571,254.09448 A 34.285713,61.42857 0 1 1 202.85714,254.09448 z" transform="translate(-53.900904,89.798016)"></path> <path style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1" d="M 110.36437,283.20569 C 627.11227,134.40393 627.11227,134.40393 627.11227,134.40393 L 628.46501,133.05119" id="path2387"></path> <path style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1" d="M 121.51454,404.3102 L 639.61519,186.51853" id="path2389"></path> <path sodipodi:type="arc" style="opacity:1;fill:#ffffff;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate" id="path2391" sodipodi:cx="168.57143" sodipodi:cy="254.09448" sodipodi:rx="34.285713" sodipodi:ry="61.42857" d="M 202.85714,254.09448 A 34.285713,61.42857 0 1 1 134.28571,254.09448 A 34.285713,61.42857 0 1 1 202.85714,254.09448 z" transform="matrix(0.8143313,0,0,0.8224064,60.410706,-1.7353843)"></path> <path transform="matrix(0.6487302,0,0,0.7455493,188.19371,-6.6397643)" d="M 202.85714,254.09448 A 34.285713,61.42857 0 1 1 134.28571,254.09448 A 34.285713,61.42857 0 1 1 202.85714,254.09448 z" sodipodi:ry="61.42857" sodipodi:rx="34.285713" sodipodi:cy="254.09448" sodipodi:cx="168.57143" id="path3163" style="opacity:1;fill:#fffdfe;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate" sodipodi:type="arc"></path> <path sodipodi:type="arc" style="opacity:1;fill:#fffdfe;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate" id="path3165" sodipodi:cx="168.57143" sodipodi:cy="254.09448" sodipodi:rx="34.285713" sodipodi:ry="61.42857" d="M 202.85714,254.09448 A 34.285713,61.42857 0 1 1 134.28571,254.09448 A 34.285713,61.42857 0 1 1 202.85714,254.09448 z" transform="matrix(0.676825,0,0,0.6872657,267.99957,-12.600074)"></path> <path transform="matrix(0.614871,0,0,-0.6371152,352.77903,305.95506)" d="M 202.85714,254.09448 A 34.285713,61.42857 0 1 1 134.28571,254.09448 A 34.285713,61.42857 0 1 1 202.85714,254.09448 z" sodipodi:ry="61.42857" sodipodi:rx="34.285713" sodipodi:cy="254.09448" sodipodi:cx="168.57143" id="path3167" style="opacity:1;fill:#fffdfe;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate" sodipodi:type="arc"></path> <path style="fill:none;fill-rule:evenodd;stroke:#000000;stroke-width:1.04990613px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1" d="M 276.5261,236.44499 C 437.3288,335.40514 437.3288,335.40514 437.3288,335.40514" id="path3211" sodipodi:nodetypes="cc"></path> <text xml:space="preserve" style="font-size:16px;font-style:normal;font-variant:normal;font-weight:normal;font-stretch:normal;text-align:start;line-height:125%;writing-mode:lr-tb;text-anchor:start;fill:#000000;fill-opacity:1;stroke:none;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;font-family:Bitstream Vera Sans;-inkscape-font-specification:Bitstream Vera Sans" x="412.48389" y="351.69116" id="text3215" sodipodi:linespacing="125%"><tspan sodipodi:role="line" id="tspan3217" x="412.48389" y="351.69116">base ray</tspan></text> <path transform="matrix(0.5637487,0,0,-0.5562563,430.14511,269.36854)" d="M 202.85714,254.09448 A 34.285713,61.42857 0 1 1 134.28571,254.09448 A 34.285713,61.42857 0 1 1 202.85714,254.09448 z" sodipodi:ry="61.42857" sodipodi:rx="34.285713" sodipodi:cy="254.09448" sodipodi:cx="168.57143" id="path2450" style="opacity:1;fill:#fffdfe;fill-opacity:1;fill-rule:evenodd;stroke:#000000;stroke-width:1px;stroke-linecap:butt;stroke-linejoin:miter;marker:none;marker-start:none;marker-mid:none;marker-end:none;stroke-miterlimit:4;stroke-dasharray:none;stroke-dashoffset:0;stroke-opacity:1;visibility:visible;display:inline;overflow:visible;enable-background:accumulate" sodipodi:type="arc"></path> </g> </svg> <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 an infinite cylinder with an infinite string of circles attached via a proper ray <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\alpha: [0,\infty) \to X</annotation></semantics></math>. The space has just one ‘end’ but you can choose different ways of going from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>x</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(U_{i+1},x_{i+1})</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(U_i,x_i)</annotation></semantics></math> for fairly obvious choices of base points such that the limit groups of the resulting two inverse systems are non-isomorphic! (In the survey listed below, this example is examined in detail, and one of the limits is a free group on one element, the other is trivial! Definitely non-isomorphic!)</p> <p>This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">lim \pi_1(\varepsilon(X))</annotation></semantics></math> is not an invariant of the end. This phenomenon is linked to the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(\varepsilon(X))</annotation></semantics></math> does not satisfy the <a class="existingWikiWord" href="/nlab/show/Mittag-Leffler+condition">Mittag-Leffler condition</a> for either choice of the base rays.</p> <h3 id="waldhausen_boundary">Waldhausen boundary</h3> <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> 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> are locally compact Hausdorff spaces, there is no obvious candidate for a space of proper maps from <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 <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>, but one can form a simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℙ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{P}(X,Y)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℙ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mo>=</mo><mi>Proper</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{P}(X,Y)_n = Proper(X\times \Delta^n,Y)</annotation></semantics></math>, which acts as if it was the <em>singular complex of the mythical space of proper maps from <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 <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></em>.</p> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>The <strong>Waldhausen boundary</strong> 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> is the simplicial set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℙ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{P}([0,\infty),X)</annotation></semantics></math>.</p> </div> <p>There is an epimorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>ℙ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\mathbb{P}([0,\infty),X))</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e(X)</annotation></semantics></math>.</p> <ul> <li> <p>In the example above of the cylinder with the string of circles attached, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>ℙ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\mathbb{P}([0,\infty),X))</annotation></semantics></math>, is uncountable, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ℙ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(\mathbb{P}([0,\infty),X))</annotation></semantics></math> maps onto <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lim</mi><mi>S</mi></mrow><annotation encoding="application/x-tex">lim S</annotation></semantics></math>.</p> </li> <li> <p>When <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> has a single end and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>ε</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\varepsilon(X))</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Mittag-Leffler+condition">Mittag-Leffler</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>ℙ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\mathbb{P}([0,\infty),X))</annotation></semantics></math>, is a single point, i.e. all possible base rays are properly homotopic.</p> </li> </ul> <h3 id="second_attempt">Second attempt</h3> <p>Even if we did not have the above difficulty with the limit groups, we would still have the problem that, as the limit functor is not exact, the resulting limiting homotopy groups would not be that well behaved. There would not be any general long exact sequence results (just as with <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+homology">Čech homology</a>). There is at least one possible replacement for those limiting homotopy groups, but first we note that it is not appropriate to base any such things at a point, rather we should be using a base ray as was discussed above.</p> <p>One fairly obvious attempt to define a ‘fundamental group’ 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>, based at a proper ray <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\alpha: [0,\infty) \to X</annotation></semantics></math>, would be to note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> gives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℙ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{P}([0,\infty),X)</annotation></semantics></math> a base point so we could look at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ℙ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(\mathbb{P}([0,\infty),X),\alpha)</annotation></semantics></math> and more generally at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>ℙ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_n(\mathbb{P}([0,\infty),X),\alpha)</annotation></semantics></math>, and we will denote these groups by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><munder><munder><mi>π</mi><mo>̲</mo></munder><mo>̲</mo></munder> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{\underline{\pi}}_n(X,\alpha)</annotation></semantics></math>.</p> <p>There are variants ‘at infinity’ of both the Waldhausen boundary and these groups, obtained using germs instead of proper maps. These will be denoted with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math> as a super- or suffix on the above notation.</p> <h3 id="browngrossman_homotopy_groups">Brown–Grossman homotopy groups</h3> <p>(Once over lightly here, more details at <a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a>.) In fact there is another different way of looking at these groups, which has a more geometric feel to it. Historically these groups were not the first successful attempt. This was due to Ed Brown and uses strings of spheres. These are examples of <a class="existingWikiWord" href="/nlab/show/spherical+objects">spherical objects</a> and the resulting ‘groups’ (better thought of as ‘<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mi>𝒜</mi></msub></mrow><annotation encoding="application/x-tex">\Pi_\mathcal{A}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">algebras</a>’) have a rich structure. They are discussed at <a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a>s. They do link with the above homotopy groups of the Waldhausen boundary, which are called the <span class="newWikiWord">Steenrod homotopy group<a href="/nlab/new/Steenrod+homotopy+group">?</a></span>s, (for reasons that will be explained there).</p> <h2 id="to_do">To do!</h2> <ul> <li>Add stuff on Steenrod homotopy groups both for here and for strong shape theory entries.</li> </ul> <h2 id="references">References</h2> <p>Survey article:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tim+Porter">Tim Porter</a>, <em>Proper Homotopy Theory</em>, in the <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em>, Ed. I.M.James, Elsevier, 1995, p. 127-167,</li> </ul> <p>Books:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hans-Joachim+Baues">H. J. Baues</a> and A. Quintero, <em>Infinite homotopy theory</em>, Volume 6 of K-monographs in mathematics, Springer, 2001</p> </li> <li> <p>Bruce Hughes and Andrew Ranicki, <em>Ends of Complexes</em>, Cambridge Tracts in Mathematics (No. 123), C.U.P.</p> </li> </ul> <p>Lecture Notes:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/D.+A.+Edwards">D. A. Edwards</a> and <a class="existingWikiWord" href="/nlab/show/H.+M.+Hastings">H. M. Hastings</a>, <em>Čech and Steenrod Homotopy Theory with Applications,</em> SLNM 542, Springer (1976).</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 28, 2018 at 07:40:27. See the <a href="/nlab/history/proper+homotopy+theory" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/proper+homotopy+theory" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><span class="backintime"><a href="/nlab/revision/proper+homotopy+theory/42" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/proper+homotopy+theory" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/proper+homotopy+theory" accesskey="S" class="navlink" id="history" rel="nofollow">History (42 revisions)</a> <a href="/nlab/show/proper+homotopy+theory/cite" style="color: black">Cite</a> <a href="/nlab/print/proper+homotopy+theory" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/proper+homotopy+theory" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>