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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="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></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='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#maximal_tori'>Maximal tori</a></li> <li><a href='#AbelianCompactLieGroups'>Abelian compact Lie groups</a></li> <li><a href='#InvariantMetric'>Invariant metric</a></li> <li><a href='#smooth_actions'>Smooth actions</a></li> <li><a href='#EquivariantTriangulationTheorem'>Equivariant triangulation theorem</a></li> <li><a href='#spaces_of_homomorphisms'>Spaces of homomorphisms</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#applications'>Applications</a></li> <ul> <li><a href='#InEquivariantHomotopyTheory'>In equivariant homotopy theory</a></li> <li><a href='#in_gauge_theory'>In gauge theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>A real <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> is <strong>compact</strong> if its underlying <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> is a <a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>.</p> <h2 id="properties">Properties</h2> <h3 id="maximal_tori">Maximal tori</h3> <p>All <a class="existingWikiWord" href="/nlab/show/maximal+torus">maximal tori</a> of a compact Lie group are conjugate by <a class="existingWikiWord" href="/nlab/show/inner+automorphisms">inner automorphisms</a>. The <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of a maximal torus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> of a compact Lie group is called the <em><a class="existingWikiWord" href="/nlab/show/rank+of+a+Lie+group">rank</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/normalizer">normalizer</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(T)</annotation></semantics></math> of a maximal torus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> determines <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Weyl+group">Weyl group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mi>W</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(G)=W(G,T)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> with respect to a choice of a maximal torus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/group+of+automorphisms">group of automorphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> which are restrictions of <a class="existingWikiWord" href="/nlab/show/inner+automorphisms">inner automorphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>The maximal torus is of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/index+of+a+subgroup">index</a> in its normalizer; the <a class="existingWikiWord" href="/nlab/show/quotient+group">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">N(T)/T</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(G)</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W(G)</annotation></semantics></math> for a compact <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, equals the <a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a> of the <a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">G/T</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/flag+variety">flag variety</a>”).</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/relation+between+compact+Lie+groups+and+reductive+algebraic+groups">relation between compact Lie groups and reductive algebraic groups</a></em></p> <h3 id="AbelianCompactLieGroups">Abelian compact Lie groups</h3> <p> <div class='num_prop' id='MaximalToriOfConnectedCompactLieGroupsAreMaximalAbelianSubgroups'> <h6>Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/maximal+torus">maximal torus</a> of a <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> is also the maximal <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a>.</p> </div> (e.g. <a href="#Salamon21">Salamon 2021, Lem. 6.5</a>)</p> <p>In particular: <div class='num_prop' id='ConnectedCompactLieGroupsAreTori'> <h6>Proposition</h6> <p>All connected compact abelian Lie groups are <a class="existingWikiWord" href="/nlab/show/tori">tori</a>, up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> (<a href="#Adams82">Adams 1982, Thm. 2.19, Cor. 2.20</a>)</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p><strong>(Classification of compact abelian Lie groups)</strong> <br /> Assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>, every <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/direct+product+group">direct product group</a> of an <a class="existingWikiWord" href="/nlab/show/n-torus">n-torus</a> with a <a class="existingWikiWord" href="/nlab/show/finite+group">finite</a> <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>.</p> </div> <div class='proof'> <h6>Proof</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi mathvariant="normal">e</mi></msub><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>G</mi></mrow><annotation encoding="application/x-tex"> G_{\mathrm{e}} \xhookrightarrow{\;\;} G </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> which is the <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> of the <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> in the given compact abelian Lie group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. By Prop. <a class="maruku-ref" href="#ConnectedCompactLieGroupsAreTori"></a> this is a <a class="existingWikiWord" href="/nlab/show/torus">torus</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi mathvariant="normal">e</mi></msub><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><msup><mi>T</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> G_{\mathrm{e}} \,\simeq\, T^n </annotation></semantics></math></div> <p>hence its <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> is a <a class="existingWikiWord" href="/nlab/show/divisible+group">divisible group</a> and therefore, by <a href="injective+object#InjectiveAbelianGroupIsDivisibleGroup">this Prop.</a>, an <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>. 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xlink:href="#2_Ogj5bOybJWm2Tx2BTB3bceQMo=-glyph-2-1" x="7.749" y="47.756"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-dasharray="3.34735 1.91277" stroke-miterlimit="10" d="M -30.507781 -23.185469 L 27.464875 21.037187 " transform="matrix(1, 0, 0, -1, 57.832, 44.795)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.487472 2.86964 C -2.032742 1.147041 -1.020532 0.335592 0.000241009 -0.000889788 C -1.01974 -0.333218 -2.033492 -1.146886 -2.487425 -2.869101 " transform="matrix(0.79504, -0.60648, -0.60648, -0.79504, 85.48755, 23.61272)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#2_Ogj5bOybJWm2Tx2BTB3bceQMo=-glyph-2-2" x="47.656" y="40.271"></use> </g> </svg> <p>hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/retraction">retracts</a> onto <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">G_0</annotation></semantics></math>.</p> <p>While this is, a priori, a diagram in abelian <a class="existingWikiWord" href="/nlab/show/discrete+groups">discrete groups</a> not it abelian Lie group, the fact that the dashed morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><msub><mi>G</mi> <mi mathvariant="normal">e</mi></msub></mrow><annotation encoding="application/x-tex">p \colon G \to G_{\mathrm{e}}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/restriction">restricts</a> to the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi mathvariant="normal">e</mi></msub></mrow><annotation encoding="application/x-tex">G_{\mathrm{e}}</annotation></semantics></math>, together with the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of copies of this connected component and using the <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> property implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> homomorphism. 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G_{\mathrm{e}} \times G/G_{\mathrm{e}} \;\simeq\; T^n \times A \,. </annotation></semantics></math></div> <p>By the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> was compact abelian, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mi>G</mi><mo stretchy="false">/</mo><msub><mi>G</mi> <mi mathvariant="normal">e</mi></msub></mrow><annotation encoding="application/x-tex">A \,\coloneqq\, G/G_{\mathrm{e}}</annotation></semantics></math> is finite abelian.</p> </div> </p> <h3 id="InvariantMetric">Invariant metric</h3> <p> <div class='num_prop' id='CompactLieGroupsAdmitBiinvariantMetrics'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a> admit <a class="existingWikiWord" href="/nlab/show/bi-invariant+Riemannian+metrics">bi-invariant Riemannian metrics</a>)</strong><br /> Every compact Lie group admits a <a class="existingWikiWord" href="/nlab/show/bi-invariant+Riemannian+metric">bi-invariant Riemannian metric</a>.</p> </div> </p> <p>(<a href="invariant+metric#Milnor76">Milnor 76, Cor. 1.4</a>)</p> <h3 id="smooth_actions">Smooth actions</h3> <div class="num_prop" id="SmoothActionOfCompactLieGroupOnSmoothManifoldIsProper"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a>. Then every <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth</a> <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/proper+action">proper</a>.</p> </div> <p>(e.g. <a href="#Lee12">Lee 12, Corollary 21.6</a>)</p> <h3 id="EquivariantTriangulationTheorem">Equivariant triangulation theorem</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/equivariant+triangulation+theorem">equivariant triangulation theorem</a></em> (<a href="equivariant+triangulation+theorem#Illman78">Illman 78</a>, <a href="equivariant+triangulation+theorem#Illman83">Illman 83</a>) says that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a compact Lie group and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> equipped with a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a>, there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+triangulation">equivariant triangulation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <h3 id="spaces_of_homomorphisms">Spaces of homomorphisms</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/nearby+homomorphisms+from+compact+Lie+groups+are+conjugate">nearby homomorphisms from compact Lie groups are conjugate</a></li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> is a compact Lie group iff it is a <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/classical+Lie+groups">classical Lie groups</a> for <em>definite</em> <a class="existingWikiWord" href="/nlab/show/inner+product+space">inner products</a> are compact, such as the <a class="existingWikiWord" href="/nlab/show/orthogonal+groups">orthogonal groups</a>, the <a class="existingWikiWord" href="/nlab/show/unitary+groups">unitary groups</a>, the <a class="existingWikiWord" href="/nlab/show/quaternionic+unitary+groups">quaternionic unitary groups</a>, etc., but not the <a class="existingWikiWord" href="/nlab/show/Lorentz+group">Lorentz group</a> etc.</p> </li> </ul> <h2 id="applications">Applications</h2> <h3 id="InEquivariantHomotopyTheory">In equivariant homotopy theory</h3> <p>Compact Lie groups make a somewhat unexpected appearance as <a class="existingWikiWord" href="/nlab/show/equivariance+groups">equivariance groups</a> in <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a>, where the compact Lie condition on the <a class="existingWikiWord" href="/nlab/show/equivariance+group">equivariance group</a> is needed in order for (the available proofs of) the <a class="existingWikiWord" href="/nlab/show/equivariant+Whitehead+theorem">equivariant Whitehead theorem</a> to hold.</p> <p>(Namely, the <a class="existingWikiWord" href="/nlab/show/equivariant+triangulation+theorem">equivariant triangulation theorem</a> <a href="#EquivariantTriangulationTheorem">above</a> is used in these proofs to guaratee that <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> of <a class="existingWikiWord" href="/nlab/show/coset+spaces">coset spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G/H</annotation></semantics></math> are themselves <a class="existingWikiWord" href="/nlab/show/G-CW-complexes">G-CW-complexes</a>.)</p> <h3 id="in_gauge_theory">In gauge theory</h3> <p>In <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>/<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>, …) …</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subgroup">closed subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+Lie+groupoid">proper Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/p-compact+group">p-compact group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+ad-equivariant+K-theory">twisted ad-equivariant K-theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>Textbook accounts:</p> <ul> <li id="Bredon72"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, Section 0.6 of: <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+compact+transformation+groups">Introduction to compact transformation groups</a></em>, Academic Press 1972 (<a href="https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1">ISBN 9780080873596</a>, <a href="http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf">pdf</a>)</p> </li> <li id="Adams82"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, <em>Lectures on Lie groups</em>, University of Chicago Press, 1982 (<a href="https://press.uchicago.edu/ucp/books/book/chicago/L/bo3614673.html">ISBN:9780226005300</a>, <a href="https://www.google.com/books/edition/Lectures_on_Lie_Groups/TC7d3ZcqjfsC?hl=en">gbooks</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hans+Duistermaat">Hans Duistermaat</a>, <a class="existingWikiWord" href="/nlab/show/Johan+A.+C.+Kolk">Johan A. C. Kolk</a>, Chapter 3 of: <em>Lie groups</em>, Springer (2000) [<a href="https://doi.org/10.1007/978-3-642-56936-4">doi:10.1007/978-3-642-56936-4</a>]</p> </li> </ul> <p>In the broader context of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>:</p> <ul> <li id="Lee12"><a class="existingWikiWord" href="/nlab/show/John+Lee">John Lee</a>, <em>Introduction to Smooth Manifolds</em>, Springer 2012 (<a href="https://doi.org/10.1007/978-1-4419-9982-5">doi:10.1007/978-1-4419-9982-5</a>, <a href="https://lost-contact.mit.edu/afs/adrake.org/usr/rkh/Books/books/Introduction%20to%20Smooth%20Manifolds%20-%20J.%20Lee.pdf">Draft pdf of the 1st edition</a>)</li> </ul> <p>Dedicated lecture notes:</p> <ul> <li id="Salamon21"><a class="existingWikiWord" href="/nlab/show/Dietmar+Salamon">Dietmar Salamon</a>, <em>Notes on compact Lie groups</em>, 2021 (<a href="https://people.math.ethz.ch/~salamon/PREPRINTS/liegroup.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Salamon_CompactLieGroups.pdf" title="pdf">pdf</a>)</li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <a class="existingWikiWord" href="/nlab/show/Theodor+Br%C3%B6cker">Theodor Bröcker</a>, <em>Representations of compact Lie groups</em>, Springer 1985 (<a href="https://link.springer.com/book/10.1007/978-3-662-12918-0">doi:10.1007/978-3-662-12918-0</a>)</li> </ul> <p>For more see also the references at <em><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on March 20, 2023 at 14:07:12. 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