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> equivariant K-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> equivariant K-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/discussion/5146/#Item_27" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <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="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="representation_theory">Representation theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+object">equivariant object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/socle">socle</a>, <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+algebra">module algebra</a>, <a class="existingWikiWord" href="/nlab/show/comodule+algebra">comodule algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+action">Hopf action</a>, <a class="existingWikiWord" href="/nlab/show/measuring">measuring</a></p> </li> </ul> <h2 id="geometric_representation_theory">Geometric representation theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/perverse+sheaf">perverse sheaf</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, <a class="existingWikiWord" href="/nlab/show/lambda-ring">lambda-ring</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+function">symmetric function</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>, <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>, <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstruction+theorems">reconstruction theorems</a></p> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Beilinson-Bernstein+localization">Beilinson-Bernstein localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/representation+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#BottPeriodicity'>Bott periodicity</a></li> <li><a href='#complex_orientation'>Complex orientation</a></li> <li><a href='#relation_to_operator_ktheory_of_crossed_product_algebras'>Relation to operator K-theory of crossed product algebras</a></li> <li><a href='#RelationToRepresentationTheory'>Relation to representation theory</a></li> <ul> <li><a href='#EquivariantKUAndTheComplexRepresentationRing'>Equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KU</mi></mrow><annotation encoding="application/x-tex">KU</annotation></semantics></math> and the complex representation ring</a></li> <li><a href='#ChernClassesOfLinearRepresentations'>Chern classes of linear representations</a></li> <li><a href='#equivariant__and_the_real_representation_ring'>Equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KO</mi></mrow><annotation encoding="application/x-tex">KO</annotation></semantics></math> and the real representation ring</a></li> </ul> <li><a href='#relation_to_ktheory_of_homotopy_quotient_spaces_borel_constructions'>Relation to K-theory of homotopy quotient spaces (Borel constructions)</a></li> <li><a href='#Rationalization'>Rationalization</a></li> <li><a href='#equivariant_cherncharacter'>Equivariant Chern-character</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesRepresentingSpectrum'>Representing equivariant spectrum</a></li> <li><a href='#ReferencesRationalEquivariantKTheory'>Rational equivariant K-theory</a></li> <li><a href='#for_dbrane_charge_on_orbifolds'>For D-brane charge on orbifolds</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Equivariant K-theory</em> is the <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> version of the <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>.</p> <p>To the extent that <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> is given by <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundles">virtual vector bundles</a> (<a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>, <a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a>), equivariant K-theory is given by equivalence classes of virtual <a class="existingWikiWord" href="/nlab/show/equivariant+bundles">equivariant bundles</a> or generalizations to <a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a> thereof, as in <em><a class="existingWikiWord" href="/nlab/show/equivariant+operator+K-theory">equivariant operator K-theory</a></em>, <em><a class="existingWikiWord" href="/nlab/show/equivariant+KK-theory">equivariant KK-theory</a></em>.</p> <h2 id="properties">Properties</h2> <h3 id="BottPeriodicity">Bott periodicity</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a> of plain K-theory generalizes to equivariant K-theory:</p> <p>Complex equivariant K-theory is invariant under smashing with <a class="existingWikiWord" href="/nlab/show/representation+spheres">representation spheres</a> of complex representations (<a href="#Atiyah68">Atiyah 68, Theorem 4.3</a>), while real equivariant K-theory is invariant under smashing with representation spheres of real 8d reps with spin structure (<a href="#Atiyah68">Atiyah 68, Theorem 6.1</a>).</p> <p>Review in <a href="#Karoubi05">Karoubi 05, Section 5</a>.</p> <h3 id="complex_orientation">Complex orientation</h3> <p>Equivariant complex K-theory is an <a class="existingWikiWord" href="/nlab/show/equivariant+complex+oriented+cohomology+theory">equivariant complex oriented cohomology theory</a> (<a href="#Greenlees01">Greenlees 01, Sec. 10</a>).</p> <div class="num_prop" id="EquivariantKTheoryOfProjectiveGSpace"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/equivariant+K-theory+of+projective+G-space">equivariant K-theory of projective G-space</a>)</strong></p> <p>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> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>⊕</mo><mi>i</mi></munder><mspace width="thinmathspace"></mspace><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>G</mi><msubsup><mi>Representations</mi> <mi>ℂ</mi> <mi>fin</mi></msubsup></mrow><annotation encoding="application/x-tex"> \underset{i}{\oplus} \, \mathbf{1}_{V_i} \;\; \in \;\; G Representations_{\mathbb{C}}^{fin} </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite-dimensional</a> <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex</a> 1-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a>.</p> <p>The <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+K-theory">equivariant K-theory</a> <a class="existingWikiWord" href="/nlab/show/cohomology+ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K_G(-)</annotation></semantics></math> of the corresponding <a class="existingWikiWord" href="/nlab/show/projective+G-space">projective G-space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(-)</annotation></semantics></math> is the following <a class="existingWikiWord" href="/nlab/show/quotient+ring">quotient ring</a> of the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> in one <a class="existingWikiWord" href="/nlab/show/variable">variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> over the complex <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(G)</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>:</p> <div class="maruku-equation" id="eq:EquivariantKTheoryRingOfProjectiveGSpace"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>G</mi></msub><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>P</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><mo>⊕</mo><mi>i</mi></munder><mspace width="thinmathspace"></mspace><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>L</mi><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>i</mi></munder><mo maxsize="1.2em" minsize="1.2em">(</mo><mn>1</mn><mo>−</mo><msub><mn>1</mn> <mrow><msub><mrow></mrow> <mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow></msub></mrow></msub><mi>L</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> K_G \Big( P \big( \underset{i}{\oplus} \, \mathbf{1}_{V_i} \big) \Big) \;\; \simeq \;\; R(G) \big[ L \big] \big/ \underset{i}{\prod} \big( 1 - 1_{{}_{V_i}} L \big) \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mi>ℒ</mi> <mrow><munder><mo>⊕</mo><mi>i</mi></munder><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow></msub></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex">L \,=\, \big[ \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} }\big]</annotation></semantics></math> is the K-theory class of the <a class="existingWikiWord" href="/nlab/show/tautological+equivariant+line+bundle">tautological equivariant line bundle</a> on the given <a class="existingWikiWord" href="/nlab/show/projective+G-space">projective G-space</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mrow><msub><mrow></mrow> <mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow></msub></mrow></msub><mi>L</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">[</mo><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow></msub><mo>⊠</mo><msub><mi>ℒ</mi> <mrow><munder><mo>⊕</mo><mi>i</mi></munder><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow></msub></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo></mrow><annotation encoding="application/x-tex"> 1_{{}_{V_i}} L \;=\; \big[ \mathbf{1}_{V_i} \boxtimes \mathcal{L}_{ \underset{i}{\oplus} \mathbf{1}_{V_i} } \big]</annotation></semantics></math> is the class of its <a class="existingWikiWord" href="/nlab/show/external+tensor+product">external tensor product</a> of <a class="existingWikiWord" href="/nlab/show/equivariant+vector+bundles">equivariant vector bundles</a> with the given <a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a>.</p> </li> </ul> </div> <p>(<a href="#Greenlees01">Greenlees 01, p. 248 (24 of 39)</a>)</p> <div class="num_cor" id="EquivariantComplexOrientationOfEquivariantComplexKTheory"> <h6 id="corollary">Corollary</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/equivariant+complex+oriented+cohomology">equivariant complex orientation</a> of <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a>)</strong></p> <p>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> an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mi>V</mi></msub><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>G</mi><msubsup><mi>Representations</mi> <mi>ℂ</mi> <mi>fin</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathbf{1}_V \,\in\, G Representations_{\mathbb{C}}^{fin}</annotation></semantics></math> a complex 1-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a>, the corresponding <a class="existingWikiWord" href="/nlab/show/representation+sphere">representation sphere</a> is the <a class="existingWikiWord" href="/nlab/show/projective+G-space">projective G-space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mi>V</mi></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><mi>P</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mi>V</mi></msub><mo>⊕</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">S^{\mathbf{1}_V} \,\simeq\, P\big( \mathbf{1}_V \oplus \mathbf{1} \big)</annotation></semantics></math> (<a href="representation+sphere#OneDimensionalRepresentationSpheresAsProjectiveLine">this Prop.</a>) and so, by Prop. <a class="maruku-ref" href="#EquivariantKTheoryOfProjectiveGSpace"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mover><mi>K</mi><mo>˜</mo></mover> <mi>G</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mrow><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mi>V</mi></msub></mrow></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mo>≃</mo><mspace width="thinmathspace"></mspace><msub><mi>K</mi> <mi>G</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>P</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mi>V</mi></msub><mo>⊕</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo stretchy="false">)</mo><mo>;</mo><mspace width="thinmathspace"></mspace><munder><munder><mrow><mi>P</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><mo>≃</mo><mspace width="thinmathspace"></mspace><mi>pt</mi></mrow></munder><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>ker</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>L</mi><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mn>1</mn> <mrow><msub><mrow></mrow> <mi>V</mi></msub></mrow></msub><mi>L</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>L</mi><mo stretchy="false">)</mo><mo>⟶</mo><munder><munder><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>L</mi><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><mo>≃</mo><mspace width="thinmathspace"></mspace><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></munder><mo maxsize="1.8em" minsize="1.8em">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>L</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>L</mi><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.2em" minsize="1.2em">/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mn>1</mn> <mrow><msub><mrow></mrow> <mi>V</mi></msub></mrow></msub><mi>L</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>L</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \widetilde K_G \big( S^{\mathbf{1}_V} \big) & \simeq\, K_G \big( P( \mathbf{1}_V \oplus \mathbf{1} ) ; \, \underset{ \simeq \, pt }{ \underbrace{ P( \mathbf{1} ) } } \big) \\ & \simeq ker \Big( R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \longrightarrow \underset{ \simeq \, R(G) }{ \underbrace{ R(G)\big[L\big] \big/ (1 - L) } } \Big) \\ & \simeq (1 - L) \cdot R(G)\big[L\big] \big/ (1 - 1_{{}_{V}} L) (1 - L) \end{aligned} </annotation></semantics></math></div> <p>is generated by the <a class="existingWikiWord" href="/nlab/show/Bott+element">Bott element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1 - L)</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mstyle mathvariant="bold"><mn>1</mn></mstyle> <mi>V</mi></msub><mo>⊕</mo><mstyle mathvariant="bold"><mn>1</mn></mstyle><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">P\big( \mathbf{1}_V \oplus \mathbf{1} \big)</annotation></semantics></math>. By the nature of the <a class="existingWikiWord" href="/nlab/show/tautological+equivariant+line+bundle">tautological equivariant line bundle</a>, this Bott element is the restriction of that on <a class="existingWikiWord" href="/nlab/show/infinite+complex+projective+G-space">infinite complex projective G-space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>𝒰</mi> <mi>G</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">P\big(\mathcal{U}_G\big)</annotation></semantics></math>. The latter is thereby exhibited as an <a class="existingWikiWord" href="/nlab/show/equivariant+complex+oriented+cohomology+theory">equivariant complex orientation</a> in <a class="existingWikiWord" href="/nlab/show/equivariant+complex+K-theory">equivariant complex K-theory</a>.</p> </div> <p>(<a href="#Greenlees01">Greenlees 01, p. 248 (24 of 39)</a>)</p> <h3 id="relation_to_operator_ktheory_of_crossed_product_algebras">Relation to operator K-theory of crossed product algebras</h3> <p>The <em><a class="existingWikiWord" href="/nlab/show/Green-Julg+theorem">Green-Julg theorem</a></em> identifies, under some conditions, equivariant K-theory with <a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a> of corresponding <a class="existingWikiWord" href="/nlab/show/crossed+product+algebras">crossed product algebras</a>.</p> <h3 id="RelationToRepresentationTheory">Relation to representation theory</h3> <h4 id="EquivariantKUAndTheComplexRepresentationRing">Equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KU</mi></mrow><annotation encoding="application/x-tex">KU</annotation></semantics></math> and the complex representation ring</h4> <p>The <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</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> over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> is the <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+K-theory">equivariant K-theory</a> of the point, or equivalently by the <a class="existingWikiWord" href="/nlab/show/Green-Julg+theorem">Green-Julg theorem</a>, if <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/compact+Lie+group">compact Lie group</a>, the <a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a> of the <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a> (the <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebra">groupoid convolution algebra</a> of the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid 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> <div class="maruku-equation" id="eq:RepresentationRingAsEquivariantKTheoryOfThePoint"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>KU</mi> <mi>G</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>KK</mi><mo stretchy="false">(</mo><mi>ℂ</mi><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R_{\mathbb{C}}(G) \simeq KU^0_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G)) \,. </annotation></semantics></math></div> <p>The first <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> here follows immediately from the elementary definition of equivariant <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>, since 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+vector+bundle">equivariant vector bundle</a> over the point is manifestly just a <a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</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 a <a class="existingWikiWord" href="/nlab/show/complex+vector+space">complex vector space</a>.</p> <p>(e.g. <a href="#Greenlees05">Greenlees 05, section 3</a>, <a href="#Wilson16">Wilson 16, example 1.6 p. 3</a>)</p> <h4 id="ChernClassesOfLinearRepresentations">Chern classes of linear representations</h4> <p>Under the identification <a class="maruku-eqref" href="#eq:RepresentationRingAsEquivariantKTheoryOfThePoint">(2)</a> and the <a class="existingWikiWord" href="/nlab/show/Atiyah-Segal+completion">Atiyah-Segal completion</a> map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>KU</mi> <mi>G</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover></mover><mi>KU</mi><mo stretchy="false">(</mo><mi>BG</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> R_{\mathbb{C}}(G) \simeq KU_G^0(\ast) \overset{ \widehat{(-)} }{\longrightarrow} KU(BG) </annotation></semantics></math></div> <p>one may ask for the <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a> of the K-theory class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>V</mi><mo>^</mo></mover><mo>∈</mo><mi>KU</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widehat{V} \in KU(B G)</annotation></semantics></math> expressed in terms of the actual <a class="existingWikiWord" href="/nlab/show/character">character</a> of the <a class="existingWikiWord" href="/nlab/show/representation">representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. For more see at <em><a class="existingWikiWord" href="/nlab/show/Chern+class+of+a+linear+representation">Chern class of a linear representation</a></em>.</p> <p>There is a closed formula at least for the <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> (<a href="#Atiyah61">Atiyah 61, appendix</a>):</p> <p>For 1-dimensional representations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> their <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mover><mi>V</mi><mo>^</mo></mover><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_1(\widehat{V}) \in H^2(B G, \mathbb{Z})</annotation></semantics></math> is their image under the canonical <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> from 1-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> characters in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>Grp</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{Grp}(G,U(1))</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2_{grp}(G, \mathbb{Z})</annotation></semantics></math> and further to the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2(B G, \mathbb{Z})</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mrow><mo>(</mo><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>Grp</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msubsup><mi>H</mi> <mi>grp</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> c_1\left(\widehat{(-)}\right) \;\colon\; Hom_{Grp}(G, U(1)) \overset{\simeq}{\longrightarrow} H^2_{grp}(G,\mathbb{Z}) \overset{\simeq}{\longrightarrow} H^2(B G, \mathbb{Z}) \,. </annotation></semantics></math></div> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> their <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mover><mi>V</mi><mo>^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_1(\widehat V)</annotation></semantics></math> is the previously defined first Chern-class of the <a class="existingWikiWord" href="/nlab/show/line+bundle">line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><msup><mo>∧</mo> <mi>n</mi></msup><mi>V</mi></mrow><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{\wedge^n V}</annotation></semantics></math> corresponding to the <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>-th <a class="existingWikiWord" href="/nlab/show/exterior+power">exterior power</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mi>n</mi></msup><mi>V</mi></mrow><annotation encoding="application/x-tex">\wedge^n V</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. The latter is a 1-dimensional representation, corresponding to the <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">determinant line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>det</mi><mo stretchy="false">(</mo><mover><mi>V</mi><mo>^</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mover><mrow><msup><mo>∧</mo> <mi>n</mi></msup><mi>V</mi></mrow><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">det(\widehat{V}) = \widehat{\wedge^n V}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mover><mi>V</mi><mo>^</mo></mover><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>det</mi><mo stretchy="false">(</mo><mover><mi>V</mi><mo>^</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mover><mrow><msup><mo>∧</mo> <mi>n</mi></msup><mi>V</mi></mrow><mo>^</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> c_1(\widehat{V}) \;=\; c_1(det(\widehat{V})) \;=\; c_1( \widehat{\wedge^n V} ) \,. </annotation></semantics></math></div> <p>(<a href="#Atiyah61">Atiyah 61, appendix, item (7)</a>)</p> <p>More explicitly, via the formula for the <a class="existingWikiWord" href="/nlab/show/determinant">determinant</a> as a <a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> in <a class="existingWikiWord" href="/nlab/show/traces">traces</a> of powers (see <a href="determinant#eq:DeterminantAsPolynomialInTracesOfPowers">there</a>) this means that the first Chern class of the <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>-dimensional representation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is expressed in terms of its <a class="existingWikiWord" href="/nlab/show/character">character</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>V</mi></msub></mrow><annotation encoding="application/x-tex">\chi_V</annotation></semantics></math> as</p> <div class="maruku-equation" id="eq:FirstChernClassOfRepresentationInTermsOfTheCharacter"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>χ</mi> <mrow><mo>(</mo><msup><mo>∧</mo> <mi>n</mi></msup><mi>V</mi><mo>)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>g</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mfrac linethickness="0"><mrow><mrow><msub><mi>k</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>k</mi> <mi>n</mi></msub><mo>∈</mo><mi>ℕ</mi></mrow></mrow><mrow><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>ℓ</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mi>ℓ</mi><msub><mi>k</mi> <mi>ℓ</mi></msub><mo>=</mo><mi>n</mi></mrow></mrow></mfrac></munder><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><msub><mi>k</mi> <mi>l</mi></msub><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><msup><mi>l</mi> <mrow><msub><mi>k</mi> <mi>l</mi></msub></mrow></msup><msub><mi>k</mi> <mi>l</mi></msub><mo>!</mo></mrow></mfrac><msup><mrow><mo>(</mo><msub><mi>χ</mi> <mi>V</mi></msub><mo stretchy="false">(</mo><msup><mi>g</mi> <mi>l</mi></msup><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><msub><mi>k</mi> <mi>l</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> c_1(V) = \chi_{\left(\wedge^n V\right)} \;\colon\; g \;\mapsto\; \underset{ { k_1,\cdots, k_n \in \mathbb{N} } \atop { \underoverset{\ell = 1}{n}{\sum} \ell k_\ell = n } }{\sum} \underoverset{ l = 1 }{ n }{\prod} \frac{ (-1)^{k_l + 1} }{ l^{k_l} k_l ! } \left(\chi_V(g^l)\right)^{k_l} </annotation></semantics></math></div> <p>For example, for a representation of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> this reduces to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>χ</mi> <mrow><mi>V</mi><mo>∧</mo><mi>V</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>g</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><msup><mrow><mo>(</mo><msub><mi>χ</mi> <mi>V</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mn>2</mn></msup><mo>−</mo><msub><mi>χ</mi> <mi>V</mi></msub><mo stretchy="false">(</mo><msup><mi>g</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> c_1(V) = \chi_{V \wedge V} \;\colon\; g \;\mapsto\; \frac{1}{2} \left( \left( \chi_V(g)\right)^2 - \chi_V(g^2) \right) </annotation></semantics></math></div> <p>(see also e.g. <a href="representation+theory#tomDieck09">tom Dieck 09, p. 45</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h4 id="equivariant__and_the_real_representation_ring">Equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KO</mi></mrow><annotation encoding="application/x-tex">KO</annotation></semantics></math> and the real representation ring</h4> <p>An isomorphism analogous to <a class="maruku-eqref" href="#eq:RepresentationRingAsEquivariantKTheoryOfThePoint">(2)</a> identifies the <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>-representation ring over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> with the equivariant orthogonal <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>-theory of the point in degree 0:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msubsup><mi>KO</mi> <mi>G</mi> <mn>0</mn></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R_{\mathbb{R}}(G) \;\simeq\; KO_G^0(\ast) \,. </annotation></semantics></math></div> <p>But beware that equivariant <a class="existingWikiWord" href="/nlab/show/KO">KO</a>, even of the point, is much richer in higher degree (<a href="#Wilson16">Wilson 16, remark 3.34</a>).</p> <p id="EquivariantKOOfThePoint"> In fact, <a class="existingWikiWord" href="/nlab/show/equivariant+KO-theory">equivariant KO-theory</a> of the point subsumes the <a class="existingWikiWord" href="/nlab/show/representation+rings">representation rings</a> over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> and the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>KO</mi> <mi>G</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>7</mn></mtd></mtr> <mtr><mtd><msub><mi>R</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>R</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>6</mn></mtd></mtr> <mtr><mtd><msub><mi>R</mi> <mi>ℍ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>R</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>5</mn></mtd></mtr> <mtr><mtd><msub><mi>R</mi> <mi>ℍ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mphantom><mrow><mo stretchy="false">/</mo><msub><mi>R</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mphantom></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>4</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>3</mn></mtd></mtr> <mtr><mtd><msub><mi>R</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>R</mi> <mi>ℍ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>2</mn></mtd></mtr> <mtr><mtd><msub><mi>R</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>R</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>1</mn></mtd></mtr> <mtr><mtd><msub><mi>R</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mphantom><mrow><mo stretchy="false">/</mo><msub><mi>R</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></mphantom></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>0</mn></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> KO_G^n(\ast) \;\simeq\; \left\{ \array{ 0 &\vert& n = 7 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6 \\ R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5 \\ R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) } &\vert& n = 4 \\ 0 &\vert& n = 3 \\ R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2 \\ R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1 \\ R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0 } \right. </annotation></semantics></math></div> <p>(<a href="#Greenlees05">Greenlees 05, p. 3</a>)</p> <p>Accordingly the construction of an <a class="existingWikiWord" href="/nlab/show/index">index</a> (<a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward</a> to the point) in equivariant K-theory is a way of producing <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/representations">representations</a> from <a class="existingWikiWord" href="/nlab/show/equivariant+vector+bundles">equivariant vector bundles</a>. This method is also called <em><a class="existingWikiWord" href="/nlab/show/Dirac+induction">Dirac induction</a></em>.</p> <p>Specifically, applied to equivariant <a class="existingWikiWord" href="/nlab/show/complex+line+bundles">complex line bundles</a> on <a class="existingWikiWord" href="/nlab/show/coadjoint+orbits">coadjoint orbits</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>, this is a K-theoretic formulation of the <a class="existingWikiWord" href="/nlab/show/orbit+method">orbit method</a>.</p> <h3 id="relation_to_ktheory_of_homotopy_quotient_spaces_borel_constructions">Relation to K-theory of homotopy quotient spaces (Borel constructions)</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> equipped with 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> 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 <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X//G</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a>. A standard model for this is the <a class="existingWikiWord" href="/nlab/show/Borel+construction">Borel construction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>≃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>EG</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X//G \simeq (X \times EG)/G \,. </annotation></semantics></math></div> <p>The ordinary <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X//G</annotation></semantics></math> is also called the <em>Borel-equivariant K-theory</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>, denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>K</mi> <mi>G</mi> <mi>Bor</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K_G^{Bor}(X) \coloneqq K(X//G) \,. </annotation></semantics></math></div> <p>There is a canonical map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>K</mi> <mi>G</mi> <mi>Bor</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K_G(X) \to K_G^{Bor}(X) </annotation></semantics></math></div> <p>from the genuine equivariant K-theory to the Borel equivariant K-theory. In terms of the <a class="existingWikiWord" href="/nlab/show/Borel+construction">Borel construction</a> this is given by the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>K</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>E</mi><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>E</mi><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>K</mi> <mi>G</mi> <mi>Bor</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> K_G(X) \to K_G(X \times E G) \simeq K((X \times E G) / G ) \simeq K_G^{Bor}(X) \,, </annotation></semantics></math></div> <p>where the first map is pullback along the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>E</mi><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times E G \to X</annotation></semantics></math> and the first equivalence holds because the <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>-action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>E</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">X \times E G</annotation></semantics></math> is free.</p> <p>This map from genuine to Borel equivariant K-theory is not in general an isomorphism.</p> <p>Specifically for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the point, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K_G(\ast) \simeq R(G)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>K</mi> <mi>G</mi> <mi>Bor</mi></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K_G^{Bor}(\ast) \simeq K(B G)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</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>-<a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a>. In this case the above canonical map is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R(G) \to K(B G) \,. </annotation></semantics></math></div> <p>This is never an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, unless <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 the trivial group. But the <a class="existingWikiWord" href="/nlab/show/Atiyah-Segal+completion+theorem">Atiyah-Segal completion theorem</a> says that the map identifies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(B G)</annotation></semantics></math> as the completion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(G)</annotation></semantics></math> at the <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> of <a class="existingWikiWord" href="/nlab/show/virtual+representations">virtual representations</a> of rank 0.</p> <div> <table><thead><tr><th>(<a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a>) <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <br /> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></th><th><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> <br /> of the <a class="existingWikiWord" href="/nlab/show/point">point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">cohomology</a> <br /> of <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+ordinary+cohomology">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">HZ</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Borel+equivariant+cohomology">Borel equivariance</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>G</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KU">KU</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>KU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>R</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">KU_G(\ast) \simeq R_{\mathbb{C}}(G)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Segal+completion+theorem">Atiyah-Segal completion theorem</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>KU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><msub><mi>KU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>≃</mo><mi>KU</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+complex+cobordism+cohomology+theory">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/MU">MU</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MU_G(\ast)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/completion+theorem+for+complex+cobordism+cohomology">completion theorem for complex cobordism cohomology</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><msub><mi>MU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>≃</mo><mi>MU</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+algebraic+K-theory">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">K \mathbb{F}_p</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub><msub><mo stretchy="false">)</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>R</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Rector+completion+theorem">Rector completion theorem</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>p</mi></msub><msub><mo stretchy="false">)</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><mo stretchy="false">(</mo><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub><msub><mo stretchy="false">)</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a></mtext></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G) </annotation></semantics></math> <br /></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+stable+cohomotopy">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><msub><mi>𝔽</mi> <mn>1</mn></msub><mover><mo>≃</mo><mtext><a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a></mtext></mover></mrow><annotation encoding="application/x-tex">K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/equivariant+sphere+spectrum">S</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Burnside+ring">Burnside ring</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕊</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{S}_G(\ast) \simeq A(G)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Segal-Carlsson+completion+theorem">Segal-Carlsson completion theorem</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a></mtext></mover><msub><mi>𝕊</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><msub><mi>𝕊</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a></mtext></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi>𝕊</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G) </annotation></semantics></math> <br /></td></tr> </tbody></table> </div> <h3 id="Rationalization">Rationalization</h3> <div> <p><strong>Incarnations of <a class="existingWikiWord" href="/nlab/show/rational+equivariant+K-theory">rational equivariant K-theory</a>:</strong></p> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> theory</th><th>definition/equivalence due to</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msubsup><mi>K</mi> <mi>G</mi> <mn>0</mn></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>;</mo><mi>ℂ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\simeq K_G^0\big(X; \mathbb{C} \big) </annotation></semantics></math></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/rational+equivariant+K-theory">rational equivariant K-theory</a></strong></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msup><mi>H</mi> <mi>ev</mi></msup><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><msup><mi>X</mi> <mi>g</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo>;</mo><mi>ℂ</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> \simeq H^{ev}\Big( \big(\underset{g \in G}{\coprod} X^g\big)/G; \mathbb{C} \Big) </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/delocalized+equivariant+cohomology">delocalized equivariant cohomology</a></td><td style="text-align: left;"><a href="delocalized+equivariant+cohomology#ConnesBaum89">Baum-Connes 89, Thm. 1.19</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msubsup><mi>H</mi> <mi>CR</mi> <mi>ev</mi></msubsup><mo maxsize="1.8em" minsize="1.8em">(</mo><mo>≺</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex">\simeq H^{ev}_{CR}\Big( \prec \big( X \!\sslash\! G\big);\, \mathbb{C} \Big)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Chen-Ruan+cohomology">Chen-Ruan cohomology</a> <br /> of <a class="existingWikiWord" href="/nlab/show/global+quotient+orbifold">global quotient orbifold</a></td><td style="text-align: left;"><a href="Chen-Ruan+cohomology#ChenRuan00">Chen-Ruan 00, Sec. 3.1</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msubsup><mi>H</mi> <mi>G</mi> <mi>ev</mi></msubsup><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>X</mi><mo>;</mo><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>↦</mo><mi>ℂ</mi><mo>⊗</mo><mi>Rep</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex">\simeq H^{ev}_G\Big( X; \, \big(G/H \mapsto \mathbb{C} \otimes Rep(H)\big) \Big)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a> <br /> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a></td><td style="text-align: left;"><a href="orbifold+cohomology#Honkasalo88">Ho88 6.5</a>+<a href="orbifold+cohomology#Honkasalu90">Ho90 5.5</a>+<a href="orbifold+cohomology#Moerdijk02">Mo02 p. 18</a>, <br /> <a href="equivariant+Chern+character#MislinValette03">Mislin-Valette 03, Thm. 6.1</a>, <br /> <a href="orbifold+cohomology#SzaboValentino07">Szabo-Valentino 07, Sec. 4.2</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msubsup><mi>K</mi> <mi>G</mi> <mn>0</mn></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>;</mo><mi>ℂ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\simeq K_G^0\big(X; \mathbb{C} \big) </annotation></semantics></math></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/rational+equivariant+K-theory">rational equivariant K-theory</a></strong></td><td style="text-align: left;"><a href="equivariant+K-theory#LueckOliver01">Lück-Oliver 01, Thm. 5.5</a>, <br /> <a href="equivariant+Chern+character#MislinValette03">Mislin-Valette 03, Thm. 6.1</a></td></tr> </tbody></table> </div> <h3 id="equivariant_cherncharacter">Equivariant Chern-character</h3> <p>There is an <a class="existingWikiWord" href="/nlab/show/equivariant+Chern+character">equivariant Chern character</a> map from equivariant K-theory to <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational</a> <a class="existingWikiWord" href="/nlab/show/equivariant+ordinary+cohomology">equivariant ordinary cohomology</a> <a href="#Rationalization">above</a></p> <p>(e.g. <a href="#Stefanich">Stefanich</a>, <a href="#SatiSchreiber20">Sati-Schreiber 20, Sec. 3.4</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+equivariant+K-theory">rational equivariant K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+Chern+character">equivariant Chern character</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baum-Connes+conjecture">Baum-Connes conjecture</a>, <a class="existingWikiWord" href="/nlab/show/Green-Julg+theorem">Green-Julg theorem</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah-Segal+completion+theorem">Atiyah-Segal completion theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold+K-theory">orbifold K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+equivariant+K-theory">twisted equivariant K-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+ad-equivariant+K-theory">twisted ad-equivariant K-theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+algebraic+K-theory">equivariant algebraic K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/McKay+correspondence">McKay correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+ordinary+cohomology">equivariant ordinary cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+ordinary+differential+cohomology">equivariant ordinary differential cohomology</a></p> </li> </ul> <h2 id="References">References</h2> <h3 id="general">General</h3> <p>The idea of equivariant <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> and the <a class="existingWikiWord" href="/nlab/show/Atiyah-Segal+completion+theorem">Atiyah-Segal completion theorem</a> goes back to</p> <ul> <li id="Atiyah61"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>Characters and cohomology of finite groups</em>, Publications Mathématiques de l’IHÉS, Volume 9 (1961) , p. 23-64 (<a href="http://www.numdam.org/item?id=PMIHES_1961__9__23_0">numdam:PMIHES_1961__9__23_0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Friedrich+Hirzebruch">Friedrich Hirzebruch</a>, <em>Vector bundles and homogeneous spaces</em>, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 (<a class="existingWikiWord" href="/nlab/files/AtiyahHirzebruch61.pdf" title="pdf">pdf</a>)</p> </li> <li id="Segal68"> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Equivariant K-theory</em>, Inst. Hautes Etudes Sci. Publ. Math. <strong>34</strong> (1968) 129-151 [<a href="http://www.numdam.org/item/PMIHES_1968__34__129_0">numdam:PMIHES_1968__34__129_0</a>]</p> </li> <li id="SegalAtiyah69"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Equivariant K-theory and completion</em>, J. Differential Geometry 3 (1969), 1–18.</p> <p>(<a href="https://projecteuclid.org/euclid.jdg/1214428815">euclid:jdg/1214428815</a>, MR 0259946)</p> </li> </ul> <p>and for <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Robert+Thomason">Robert Thomason</a>, <em>Algebraic K-theory of group scheme actions</em>, Algebraic topology and algebraic K-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 539–563</li> </ul> <p>with construction via <a class="existingWikiWord" href="/nlab/show/permutative+categories">permutative categories</a> in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bertrand+Guillou">Bertrand Guillou</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>Equivariant iterated loop space theory and permutative G-categories</em>, <em>Algebr. Geom. Topol. 17 (2017) 3259-3339</em> (<a href="https://arxiv.org/abs/1207.3459">arXiv:1207.3459</a>)</li> </ul> <p>See also at <em><a href="algebraic+K-theory#ReferencesAlgebraicKTheoryForQuotientStacks">algebraic K-theory – References – On quotient stacks</a></em>.</p> <p>Introductions and surveys:</p> <ul> <li id="Greenlees05"> <p><a class="existingWikiWord" href="/nlab/show/John+Greenlees">John Greenlees</a>, <em>Equivariant version of real and complex connective K-theory</em>, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. (<a href="http://projecteuclid.org/euclid.hha/1139839291">Euclid:1139839291</a>)</p> </li> <li> <p>N. C. Phillips, <em>Equivariant K-theory for proper actions</em>, Pitman Research Notes in Mathematics Series 178, Longman, Harlow, UK, 1989.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bruce+Blackadar">Bruce Blackadar</a>, section 11 of <em><a class="existingWikiWord" href="/nlab/show/K-Theory+for+Operator+Algebras">K-Theory for Operator Algebras</a></em></p> </li> <li> <p>Alexander Merkujev, <em>Equivariant K-theory</em> (<a href="https://www.math.ucla.edu/~merkurev/papers/5-2.pdf">pdf</a>)</p> </li> <li> <p>Zachary Maddock, <em>An informal discourse on equivariant K-theory</em> (<a href="http://math.columbia.edu/~ellis/ass/equiv_k-thry.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>, <em><a href="https://amathew.wordpress.com/2011/12/03/equivariant-k-theory/">Equivariant K-theory</a></em></p> </li> <li id="Wilson16"> <p><a class="existingWikiWord" href="/nlab/show/Dylan+Wilson">Dylan Wilson</a>, <em>Equivariant K-theory</em>, 2016 (<a href="https://www.math.uchicago.edu/~dwilson/notes/equivariant-k-theory-talk.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/WilsonKTheory16.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li id="HJJS08"><a class="existingWikiWord" href="/nlab/show/Dale+Husem%C3%B6ller">Dale Husemöller</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Joachim">Michael Joachim</a>, <a class="existingWikiWord" href="/nlab/show/Branislav+Jur%C4%8Do">Branislav Jurčo</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Schottenloher">Martin Schottenloher</a>, Section 14 of: <em><a class="existingWikiWord" href="/nlab/show/Basic+Bundle+Theory+and+K-Cohomology+Invariants">Basic Bundle Theory and K-Cohomology Invariants</a></em>, Springer Lecture Notes in Physics <strong>726</strong>, 2008, (<a href="http://www.mathematik.uni-muenchen.de/~schotten/Texte/978-3-540-74955-4_Book_LNP726.pdf">pdf</a>, <a href="https://link.springer.com/book/10.1007/978-3-540-74956-1">doi:10.1007/978-3-540-74956-1</a>)</li> </ul> <p>See also</p> <ul> <li id="BrunerGreenlees10"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Bruner">Robert Bruner</a>, <a class="existingWikiWord" href="/nlab/show/John+Greenlees">John Greenlees</a>, <em>Connective Real K-Theory of Finite Groups</em>, Mathematical Surveys and Monographs <strong>169</strong> AMS 2010 (<a href="https://bookstore.ams.org/surv-169">ISBN:978-0-8218-5189-0</a>)</p> </li> <li id="Cantarero09"> <p><a class="existingWikiWord" href="/nlab/show/Jose+Cantarero">Jose Cantarero</a>, <em>Equivariant K-theory, groupoids and proper actions</em>, Thesis 2009 (<a href="https://open.library.ubc.ca/cIRcle/collections/ubctheses/24/items/1.0068026">ubctheses:1.0068026</a>, <a class="existingWikiWord" href="/nlab/files/CantareroEquivariantKTheory.pdf" title="pdf">pdf</a>)</p> </li> <li id="Cantarero12"> <p><a class="existingWikiWord" href="/nlab/show/Jose+Cantarero">Jose Cantarero</a>, <em>Equivariant K-theory, groupoids and proper actions</em>, Journal of K-Theory, Volume 9, Issue 3 June 2012, pp. 475 - 501 (<a href="https://arxiv.org/abs/0803.3244">arXiv:0803.3244</a>, <a href="https://doi.org/10.1017/is011011005jkt173">doi:10.1017/is011011005jkt173</a>)</p> <blockquote> <p>(short version of <a href="#Cantarero09">Cantarero 09</a>)</p> </blockquote> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a> in equivariant K-theory:</p> <ul> <li id="Atiyah68"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>Bott periodicity and the index of elliptic operators</em>, The Quarterly Journal of Mathematics, Volume 19, Issue 1, 1968, Pages 113–140 (<a href="https://doi.org/10.1093/qmath/19.1.113">doi:10.1093/qmath/19.1.113</a>)</p> </li> <li id="Karoubi05"> <p><a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <em>Bott Periodicity in Topological, Algebraic and Hermitian K-Theory</em>, In: Friedlander E., Grayson D. (eds) <a class="existingWikiWord" href="/nlab/show/Handbook+of+K-Theory">Handbook of K-Theory</a>, Springer 2005 (<a href="https://doi.org/10.1007/978-3-540-27855-9_4">doi:10.1007/978-3-540-27855-9_4</a>)</p> </li> </ul> <p>Basic computations:</p> <ul> <li id="Yang95"> <p><a class="existingWikiWord" href="/nlab/show/Yimin+Yang">Yimin Yang</a>, <em>On the Coefficient Groups of Equivariant K-Theory</em>, Transactions of the American Mathematical Society</p> <p>Vol. 347, No. 1 (Jan., 1995), pp. 77-98 (<a href="https://www.jstor.org/stable/2154789">jstor:2154789</a>)</p> </li> <li id="Karoubi02"> <p><a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <em>Equivariant K-theory of real vector spaces and real vector bundles</em>, Topology and its Applications, 122, (2002) 531-456 (<a href="https://arxiv.org/abs/math/0509497">arXiv:math/0509497</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/equivariant+Chern+character">equivariant Chern character</a> is discussed in</p> <ul> <li id="Stefanich"> <p>German Stefanich, <em>Chern Character in Twisted and Equivariant K-Theory</em> (<a href="https://math.berkeley.edu/~germans/Chern2.pdf">pdf</a>)</p> </li> <li id="SatiSchreiber20"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, Sec. 3.4 of: <em><a class="existingWikiWord" href="/schreiber/show/The+Character+Map+in+Equivariant+Twistorial+Cohomotopy">The character map in equivariant twistorial Cohomotopy</a></em> (<a href="https://arxiv.org/abs/2011.06533">arXiv:2011.06533</a>)</p> </li> </ul> <p>Discussion relating to K-theory of <a class="existingWikiWord" href="/nlab/show/homotopy+quotients">homotopy quotients</a>/<a class="existingWikiWord" href="/nlab/show/Borel+constructions">Borel constructions</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, around p. 11 of <em><a class="existingWikiWord" href="/nlab/show/A+Survey+of+Elliptic+Cohomology">A Survey of Elliptic Cohomology</a></em> (<a href="http://www.math.harvard.edu/~lurie/papers/survey.pdf">pdf</a>)</li> </ul> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/twisted+ad-equivariant+K-theory">twisted ad-equivariant K-theory</a> of <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em><a class="existingWikiWord" href="/nlab/show/Loop+Groups+and+Twisted+K-Theory">Loop Groups and Twisted K-Theory</a></em>.</li> </ul> <p>Discussion of K-theory of <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> is for instance in section 3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, Johanna Leida, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a>, <em>Orbifolds and string topology</em>, Cambridge Tracts in Mathematics 171, 2007 (<a href="http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf">pdf</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> of <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Schick">Thomas Schick</a>, <em>Differential orbifold K-Theory</em>, J. Noncommut. Geom. 7 (2013), no. 4, 1027-1104 (<a href="https://arxiv.org/abs/0905.4181">arXiv:0905.4181</a>)</li> </ul> <p>Discussion of combined <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a> and <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant</a> and <a class="existingWikiWord" href="/nlab/show/real+K-theory">real</a> K-theory</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/El-ka%C3%AFoum+M.+Moutuou">El-kaïoum M. Moutuou</a>, <em>Twistings of KR for Real groupoids</em> (<a href="http://arxiv.org/abs/1110.6836">arXiv:1110.6836</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/El-ka%C3%AFoum+M.+Moutuou">El-kaïoum M. Moutuou</a>, <em>Graded Brauer groups of a groupoid with involution</em>, J. Funct. Anal. 266 (2014), no.5 (<a href="https://arxiv.org/abs/1202.2057">arXiv:1202.2057</a>)</p> </li> <li id="Freed12"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>Lectures on twisted K-theory and orientifolds</em>, lectures at ESI Vienna, 2012 (<a class="existingWikiWord" href="/nlab/files/FreedESI2012.pdf" title="pdf">pdf</a>)</p> </li> <li id="FreedMoore13"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Gregory+Moore">Gregory Moore</a>, Section 7 of: <em>Twisted equivariant matter</em>, Ann. Henri Poincaré (2013) 14: 1927 (<a href="https://arxiv.org/abs/1208.5055">arXiv:1208.5055</a>)</p> </li> <li id="Gomi17"> <p><a class="existingWikiWord" href="/nlab/show/Kiyonori+Gomi">Kiyonori Gomi</a>, <em>Freed-Moore K-theory</em> (<a href="https://arxiv.org/abs/1705.09134">arXiv:1705.09134</a>, <a href="http://inspirehep.net/record/1601772">spire:1601772</a>)</p> </li> </ul> <p>Discussion in the context of <a class="existingWikiWord" href="/nlab/show/equivariant+complex+oriented+cohomology+theory">equivariant complex oriented cohomology theory</a>:</p> <ul> <li id="Greenlees01"><a class="existingWikiWord" href="/nlab/show/John+Greenlees">John Greenlees</a>, <em>Equivariant formal group laws and complex oriented cohomology theories</em>, Homology Homotopy Appl. Volume 3, Number 2 (2001), 225-263 (<a href="https://projecteuclid.org/euclid.hha/1139840255">euclid:hha/1139840255</a>)</li> </ul> <p>For formulation and proof of the <a class="existingWikiWord" href="/nlab/show/McKay+correspondence">McKay correspondence</a>:</p> <ul> <li id="GSV83"><a class="existingWikiWord" href="/nlab/show/G%C3%A9rard+Gonzalez-Sprinberg">Gérard Gonzalez-Sprinberg</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Louis+Verdier">Jean-Louis Verdier</a>, <em>Construction géométrique de la correspondance de McKay</em>, Ann. Sci. ́École Norm. Sup. <strong>16</strong> 3 (1983) 409–449 (<a href="http://www.numdam.org/item?id=ASENS_1983_4_16_3_409_0">numdam:ASENS_1983_4_16_3_409_0</a>)</li> </ul> <h3 id="ReferencesRepresentingSpectrum">Representing equivariant spectrum</h3> <p>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>-equivariant <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> is <a class="existingWikiWord" href="/nlab/show/representable+functor">represented</a> by a <a class="existingWikiWord" href="/nlab/show/topological+G-space">topological G-space</a> is due to:</p> <ul> <li id="Matumoto71"> <p>Takao Matumoto, <em>Equivariant K-theory and Fredholm operators</em>, J. Fac. Sci. Tokyo 18 (1971/72), 109-112 (<a href="https://repository.dl.itc.u-tokyo.ac.jp/?action=repository_action_common_download&item_id=39826&item_no=1&attribute_id=19&file_no=1">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MatumotoEquivariantKTheory.pdf" title="pdf">pdf</a>)</p> </li> <li id="AtiyahSegal04"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, Sec. 6 and Corollary A3.2 in: <em>Twisted K-theory</em>, Ukrainian Math. Bull. <strong>1</strong>, 3 (2004) (<a href="http://arxiv.org/abs/math/0407054">arXiv:math/0407054</a>, <a href="http://iamm.su/en/journals/j879/?VID=10">journal page</a>, <a href="http://iamm.su/upload/iblock/45e/t1-n3-287-330.pdf">published pdf</a>)</p> </li> <li id="LueckOliver01"> <p><a class="existingWikiWord" href="/nlab/show/Wolfgang+L%C3%BCck">Wolfgang Lück</a>, <a class="existingWikiWord" href="/nlab/show/Bob+Oliver">Bob Oliver</a>, Section 1 of: <em>Chern characters for the equivariant K-theory of proper G-CW-complexes</em>, In: Aguadé J., Broto C., Casacuberta C. (eds.) <em>Cohomological Methods in Homotopy Theory</em> Progress in Mathematics, vol 196. Birkhäuser 2001 (<a href="https://doi.org/10.1007/978-3-0348-8312-2_15">doi:10.1007/978-3-0348-8312-2_15</a>)</p> </li> </ul> <p>This is enhanced to a representing <a class="existingWikiWord" href="/nlab/show/naive+G-spectrum">naive G-spectrum</a> in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, Appendix A.5 of: <em>Loop groups and twisted K-theory I</em>, Journal of Topology, Volume 4, Issue 4, December 2011, Pages 737–798 (<a href="https://arxiv.org/abs/0711.1906">arXiv:0711.1906</a>)</li> </ul> <p>In its incarnation (under <a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a>) as a <a class="existingWikiWord" href="/nlab/show/Spectra">Spectra</a>-valued <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on the <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/orbit+category">orbit category</a> this is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/James+Davis">James Davis</a>, <a class="existingWikiWord" href="/nlab/show/Wolfgang+L%C3%BCck">Wolfgang Lück</a>, <em>Spaces over a Category and Assembly Maps in Isomorphism Conjectures in K- and L-Theory</em>, K-Theory 15:201–252, 1998 (<a href="https://jfdmath.sitehost.iu.edu/teaching/m721F19/assembly.pdf">pdf</a>)</li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Valentin+Zakharevich">Valentin Zakharevich</a>, Section 2.2 of: <em>K-Theoretic Computation of the Verlinde Ring</em>, thesis 2018 (<a href="http://hdl.handle.net/2152/67663">hdl:2152/67663</a>, <a href="http://www.math.jhu.edu/~vzakharevich/research/Dissertation.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/ZakharevichKTheoryAndVerlindeRing.pdf" title="pdf">pdf</a>)</p> </li> <li id="Ortiz09"> <p>Michael L. Ortiz, Theorem 2.2 in: <em>Differential Equivariant K-Theory</em> (<a href="https://arxiv.org/abs/0905.0476">arXiv:0905.0476</a>)</p> </li> <li> <p><a href="#Cantarero09">Cantarero 09, Thm. 1.0.2 & Sec. 3.4</a></p> </li> </ul> <h3 id="ReferencesRationalEquivariantKTheory">Rational equivariant K-theory</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/rational+equivariant+K-theory">rational equivariant K-theory</a> (see also the references at <em><a class="existingWikiWord" href="/nlab/show/equivariant+Chern+character">equivariant Chern character</a></em>):</p> <ul> <li id="ConnesBaum89"> <p><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Paul+Baum">Paul Baum</a>, <em>Chern character for discrete groups</em>, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 (<a href="https://doi.org/10.1016/B978-0-12-480440-1.50015-0">doi:10.1016/B978-0-12-480440-1.50015-0</a>)</p> </li> <li id="LueckOliver01"> <p><a class="existingWikiWord" href="/nlab/show/Wolfgang+L%C3%BCck">Wolfgang Lück</a>, <a class="existingWikiWord" href="/nlab/show/Bob+Oliver">Bob Oliver</a>, Section 1 of: <em>Chern characters for the equivariant K-theory of proper G-CW-complexes</em>, In: Aguadé J., Broto C., Casacuberta C. (eds.) <em>Cohomological Methods in Homotopy Theory</em> Progress in Mathematics, vol 196. Birkhäuser 2001 (<a href="https://doi.org/10.1007/978-3-0348-8312-2_15">doi:10.1007/978-3-0348-8312-2_15</a>)</p> </li> <li id="MislinValette03"> <p><a class="existingWikiWord" href="/nlab/show/Guido+Mislin">Guido Mislin</a>, <a class="existingWikiWord" href="/nlab/show/Alain+Valette">Alain Valette</a>, Theorem 6.1 in: <em>Proper Group Actions and the Baum-Connes Conjecture</em>, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 (<a href="https://link.springer.com/book/10.1007/978-3-0348-8089-3">doi:10.1007/978-3-0348-8089-3</a>)</p> </li> </ul> <p>and with emphasis of <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>-structure:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Anna+Marie+Bohmann">Anna Marie Bohmann</a>, <a class="existingWikiWord" href="/nlab/show/Christy+Hazel">Christy Hazel</a>, <a class="existingWikiWord" href="/nlab/show/Jocelyne+Ishak">Jocelyne Ishak</a>, <a class="existingWikiWord" href="/nlab/show/Magdalena+K%C4%99dziorek">Magdalena Kędziorek</a>, <a class="existingWikiWord" href="/nlab/show/Clover+May">Clover May</a>, <em>Naive-commutative structure on rational equivariant K-theory for abelian groups</em> (<a href="https://arxiv.org/abs/2002.01556">arXiv:2002.01556</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anna+Marie+Bohmann">Anna Marie Bohmann</a>, <a class="existingWikiWord" href="/nlab/show/Christy+Hazel">Christy Hazel</a>, <a class="existingWikiWord" href="/nlab/show/Jocelyne+Ishak">Jocelyne Ishak</a>, <a class="existingWikiWord" href="/nlab/show/Magdalena+K%C4%99dziorek">Magdalena Kędziorek</a>, <a class="existingWikiWord" href="/nlab/show/Clover+May">Clover May</a>, <em>Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups</em> (<a href="https://arxiv.org/abs/2104.01079">arXiv:2104.01079</a>)</p> </li> </ul> <h3 id="for_dbrane_charge_on_orbifolds">For D-brane charge on orbifolds</h3> <p>The proposal that <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> on <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> is given by <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a> (see at <em><a class="existingWikiWord" href="/nlab/show/D-brane+charge+quantization+in+K-theory">D-brane charge quantization in K-theory</a></em>) goes back to</p> <ul> <li id="Witten98"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, section 5.1 of <em>D-Branes And K-Theory</em>, JHEP 9812:019,1998 (<a href="http://arxiv.org/abs/hep-th/9810188">arXiv:hep-th/9810188</a>)</li> </ul> <p>but it was pointed out that only a subgroup or quotient group of equivariant K-theory can be physically relevant, in</p> <ul> <li id="BDHKMMS01"><a class="existingWikiWord" href="/nlab/show/Jan+de+Boer">Jan de Boer</a>, <a class="existingWikiWord" href="/nlab/show/Robbert+Dijkgraaf">Robbert Dijkgraaf</a>, <a class="existingWikiWord" href="/nlab/show/Kentaro+Hori">Kentaro Hori</a>, <a class="existingWikiWord" href="/nlab/show/Arjan+Keurentjes">Arjan Keurentjes</a>, <a class="existingWikiWord" href="/nlab/show/John+Morgan">John Morgan</a>, <a class="existingWikiWord" href="/nlab/show/David+Morrison">David Morrison</a>, <a class="existingWikiWord" href="/nlab/show/Savdeep+Sethi">Savdeep Sethi</a>, around (137) of <em>Triples, Fluxes, and Strings</em>, Adv.Theor.Math.Phys. 4 (2002) 995-1186 (<a href="https://arxiv.org/abs/hep-th/0103170">arXiv:hep-th/0103170</a>)</li> </ul> <p>For further references see at <em><a class="existingWikiWord" href="/nlab/show/fractional+D-brane">fractional D-brane</a></em>.</p> <p>On <a class="existingWikiWord" href="/nlab/show/Chern+classes+of+linear+representations">Chern classes of linear representations</a>:</p> <ul> <li> <p><a href="#Atiyah61">Atiyah 61, Appendix</a></p> </li> <li id="Evens1965"> <p>L. Evens, <em>On the Chern classes of representations of finite groups</em>, Trans. Am. Math. Soc. 115, 180-193 (1965) (<a href="https://www.jstor.org/stable/1994264">doi:10.2307/1994264</a>)</p> </li> <li> <p>F. Kamber, Ph. Tondeur, <em>Flat Bundles and Characteristic Classes of Group-Representations</em>, American Journal of Mathematics Vol. 89, No. 4 (Oct., 1967), pp. 857-886 (<a href="https://www.jstor.org/stable/2373408">doi:10.2307/2373408</a>)</p> </li> <li id="Symonds91"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Symonds">Peter Symonds</a>, <em>A splitting principle for group representations</em>, Comment. Math. Helv. (1991) 66: 169 (<a href="https://doi.org/10.1007/BF02566643">doi:10.1007/BF02566643</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 25, 2025 at 13:04:53. See the <a href="/nlab/history/equivariant+K-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/equivariant+K-theory" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/5146/#Item_27">Discuss</a><span class="backintime"><a href="/nlab/revision/equivariant+K-theory/69" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/equivariant+K-theory" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/equivariant+K-theory" accesskey="S" class="navlink" id="history" rel="nofollow">History (69 revisions)</a> <a href="/nlab/show/equivariant+K-theory/cite" style="color: black">Cite</a> <a href="/nlab/print/equivariant+K-theory" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/equivariant+K-theory" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>