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> generalized (Eilenberg-Steenrod) cohomology 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> generalized (Eilenberg-Steenrod) cohomology </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/6930/#Item_9" 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/%3Fech+cohomology">?ech 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="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#TheEilenbergSteenrodAxioms'>Definition</a></li> <ul> <li><a href='#ReducedCohomology'>Reduced cohomology</a></li> <li><a href='#UnreducedCohomology'>Unreduced cohomology</a></li> </ul> <li><a href='#RelationBetweenReducedAndUnreduced'>Relation between reduced and unreduced cohomology</a></li> <li><a href='#brown_functoriality'>Brown functoriality</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#expression_by_ordinary_cohomology_via_atiyahhirzebruch_spectral_sequence'>Expression by ordinary cohomology via Atiyah-Hirzebruch spectral sequence</a></li> <li><a href='#WhiteheadTheorem'>Whitehead theorem</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>The collection of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> from (<a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a>) <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> to <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> which assign <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> of <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> (e.g. <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a>) may be <a class="existingWikiWord" href="/nlab/show/axiom">axiomatized</a> by a small set of natural conditions, called the <em>Eilenberg-Steenrod axioms</em> (<a href="#EilenbergSteenrod52">Eilenberg-Steenrod 52, I.3</a>), see <a href="#TheEilenbergSteenrodAxioms">below</a>. One of these conditions, the “dimension axiom” (<a href="#EilenbergSteenrod52">Eilenberg-Steenrod 52, I.3 Axiom 7</a>) says that the (co)homology groups assigned to the point are concentrated in degree 0. The class of functors obtained by discarding this “dimension axiom” came to be known as <em>generalized (co)homology theories</em> (<a href="#Whitehead62">Whitehead 62</a>) or <em>extraordinary (co)homology theories</em>.</p> <p>Examples include <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> (<a href="https://ncatlab.org/nlab/show/topological+K-theory#AtiyahHirzebruch61">Atiyah-Hirzebruch 61, 1.8</a>), <a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a>. Dually one speaks of <em><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></em>.</p> <p>Notice that, while the terminology “generalized cohomology” is standard in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> with an eye towards <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>, it is somewhat unfortunate in that there are various <em>other</em> and <em>further</em> generalizations of the axioms that all still deserve to be and are called “<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>”. For instance dropping the <a class="existingWikiWord" href="/nlab/show/suspension+isomorphism">suspension</a> axiom leads to <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> and dropping the “homotopy axiom” (and taking the domain spaces to be <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>) leads to the further generality of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>. This entry here is concerned with the generalization obtained from the Eilenberg-Steenrod axioms by <em>just</em> discarding the dimension axiom. For lack of a better term, we say “generalized (Eilenberg-Steenrod) cohomology” here.</p> <p>In (<a href="#Whitehead62">Whitehead 62</a>) it was observed that every <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> induces a generalized homology theory. The <em><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></em> (<a href="#Brown62">Brown 62</a>) asserts that every generalized (co)homology arises this way, being <a class="existingWikiWord" href="/nlab/show/representable+functor">represented</a> by <a class="existingWikiWord" href="/nlab/show/mapping+spectra">mapping spectra</a> into/<a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> with a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a>. But beware that the homology theory represented by a spectrum in general contains strictly less information than the spectrum, due to the existence of “<a class="existingWikiWord" href="/nlab/show/phantom+maps">phantom maps</a>”.</p> <p>On the other hand, if one refines the concept of a generalized homology theory from taking values in <a class="existingWikiWord" href="/nlab/show/graded+object">graded</a> <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> to taking values in <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> then it does become equivalent to the concept of <em>spectrum</em>, this is the statement at <em><a href="excisive+%28%E2%88%9E%2C1%29-functor#SpectrumObjects">excisive functor – Examples – Spectrum objects</a></em>.</p> <p>This means that from a perspective of <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>, generalized Eilenberg-Steenrod cohomology is the intrinsic <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+spectra">(∞,1)-category of spectra</a>, or better: <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> generalized Eilenberg-Steenrod cohomology is the intrinsic cohomology of the <a class="existingWikiWord" href="/nlab/show/tangent+%28%E2%88%9E%2C1%29-topos">tangent (∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/parameterized+spectra">parameterized spectra</a>.</p> <div class="standout"> <p>Generalized Eilenberg-Steenrod cohomology is <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(X)= H(X,E)</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a>.</p> </div> <h2 id="TheEilenbergSteenrodAxioms">Definition</h2> <p>This sections states the classical formulation of the Eilenberg-Steenrod axioms due to (<a href="#EilenbergSteenrod52">Eilenberg-Steenrod 52, I.3</a>) in terms of concepts from classical <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>, such as <a class="existingWikiWord" href="/nlab/show/CW-pairs">CW-pairs</a> and <a class="existingWikiWord" href="/nlab/show/mapping+cones">mapping cones</a>.</p> <p>More abstractly, via the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>, these structures are seen to serve as presentations for certain <a class="existingWikiWord" href="/nlab/show/homotopy+pushouts">homotopy pushouts</a>. In terms of “abstract homotopy theory” (<a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+theory">(infinity,1)-category theory</a>) one obtains a more streamlined formulation, which we turn to <a href="#AbstractFormulation">below</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> <p>There are two versions of the statement of the axioms:</p> <ul> <li> <p><em><a href="#ReducedCohomology">Reduced cohomology</a></em>;</p> </li> <li> <p><em><a href="#UnreducedCohomology">Unreduced cohomology</a></em>.</p> </li> </ul> <p>There are functors taking any reduced cohomology theory to an unreduced one, and vice versa. When some fine detail in the axioms is suitably set up, then this establishes an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> between reduced and unreduced generalized cohomology:</p> <ul> <li><em><a href="#RelationBetweenReducedAndUnreduced">Relation between reduced and unreduced cohomology</a></em></li> </ul> <p>The fine detail in the axioms that makes this work is such as to ensure that a cohomology theory is a functor on the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of the (pointed/pairwise) <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a>. Since this has different presentations, there are corresponding different versions of suitable axioms:</p> <ol> <li> <p>On the one hand, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top_{Quillen})</annotation></semantics></math> may be presented by topological spaces homeomorphic to <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a> and with <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>-classes of continuous functions between them, and accordingly a generalized cohomology theory may be taken to be a funtor on (pointed/pairs of) CW-complexes invariant under <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>.</p> </li> <li> <p>On the other hand, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>Top</mi> <mi>Quillen</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Top_{Quillen})</annotation></semantics></math> may be presented by all topological spaces with <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> <a class="existingWikiWord" href="/nlab/show/localization">inverted</a>, and accordingly a generalized cohomology theory may be taken to be a functor on all (pointed/pairs of) topological spaces that sends weak homotopy equivalences to isomorphisms.</p> </li> </ol> <p>Notice however that “<a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a>” is already ambiguous. Pre Quillen this was the category of <em>all</em> topological spaces with homotopy equivalence classes of maps between them, and often generalized cohomology functors are defined on this larger category and only restricted to CW-complexes or required to preserve weak homotopy equivalences when need be (e.g. <a href="#Switzer75">Switzer 75, p.117</a>), such as for establishing the equivalence between reduced and unreduced theories.</p> <p>Moreover, historically, these conditions have been decomposed in several numbers of ways. Notably (<a href="#EilenbergSteenrod52">Eilenberg-Steenrod 52</a>) originally listed 7 axioms for unreduced cohomology, more than typically counted today, but their axioms 1 and 2 jointly just said that we have a functor on topological spaces, axiom 3 was the condition for the connecting homomorphism to be a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>, conditions which later (<a href="#Switzer75">Switzer 75, p. 99,100</a>) were absorbed in the underlying structure.</p> <p>Finally, following the historical development it is common to state the exactness properties of cohomology functors in terms of <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> constructions. These are models for <a class="existingWikiWord" href="/nlab/show/homotopy+cofibers">homotopy cofibers</a>, but in general only when some technical conditions are met, such that the underlying topological spaces are CW-complexes.</p> <p>For these reasons, in the following we stick to two points of views: where we discuss cohomology theories as functors on topological spaces we restrict attention to those homeomorphic to CW-complexes. In a second description we speak fully abstractly about functors on the homotopy category of a given model category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category.</p> <h3 id="ReducedCohomology">Reduced cohomology</h3> <p>Throughout, write <a class="existingWikiWord" href="/nlab/show/Top">Top</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>CW</mi></msub></mrow><annotation encoding="application/x-tex">{}_{CW}</annotation></semantics></math> for the category of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top^{\ast/}_{CW}</annotation></semantics></math> for the corresponding category of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a>.</p> <p>Recall (<a href="final+functor#FiberProductsInASliceCategory">here</a>) that <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Top^{\ast/}</annotation></semantics></math> are computed as colimits in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> (<a href="Top#UniversalConstructions">here</a>) after adjoining the base point and its inclusion maps to the given diagram.</p> <div class="num_example" id="WedgeSumAsCoproduct"> <h6 id="example">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> is the <em><a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a></em>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∨</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\vee_{i \in I} X_i</annotation></semantics></math>.</p> </div> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>≔</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>⟶</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> \Sigma \coloneqq S^1 \wedge (-) \;\colon\; Top^{\ast/}_{CW} \longrightarrow Top^{\ast/}_{CW} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a> functor.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex">Ab^{\mathbb{Z}}</annotation></semantics></math> for the category of integer-<a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a>.</p> <div class="num_defn" id="ReducedGeneralizedCohomology"> <h6 id="definition_2">Definition</h6> <p>A <strong>reduced <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a></strong> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> (“<a class="existingWikiWord" href="/nlab/show/pullback+in+cohomology">pullback in cohomology</a>”)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>⟶</mo><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex"> \tilde E^\bullet \;\colon\; (Top^{\ast/}_{CW})^{op} \longrightarrow Ab^{\mathbb{Z}} </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> (<a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>) to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a> (“<a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a>”), in components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \tilde E \;\colon\; (X \stackrel{f}{\longrightarrow} Y) \mapsto (\tilde E^\bullet(Y) \stackrel{f^\ast}{\longrightarrow} \tilde E^\bullet(X)) \,, </annotation></semantics></math></div> <p>and equipped with a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of degree +1, to be called the <strong><a class="existingWikiWord" href="/nlab/show/suspension+isomorphism">suspension isomorphism</a></strong>, 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>σ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo>−</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \sigma \;\colon\; \tilde E^{\bullet +1}(\Sigma -) \overset{\simeq}{\longrightarrow} \tilde E^\bullet(-) </annotation></semantics></math></div> <p id="AxiomsReduced"> such that:</p> <ol> <li> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopy+invariance">homotopy invariance</a>)</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f_1,f_2 \colon X \longrightarrow Y</annotation></semantics></math> are two morphisms of pointed topological spaces such that there is a (base point preserving) <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>≃</mo><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_1 \simeq f_2</annotation></semantics></math> between them, then the induced <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of abelian groups are <a class="existingWikiWord" href="/nlab/show/equality">equal</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>f</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>=</mo><msubsup><mi>f</mi> <mn>2</mn> <mo>*</mo></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f_1^\ast = f_2^\ast \,. </annotation></semantics></math></div></li> <li id="ReducedExactnessAxiom"> <p><strong>(exactness)</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon A \hookrightarrow X</annotation></semantics></math> an inclusion of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j \colon X \longrightarrow Cone(i)</annotation></semantics></math> the induced <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, then this gives an <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>j</mi> <mo>*</mo></msup></mrow></mover><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde E^\bullet(Cone(i)) \overset{j^\ast}{\longrightarrow} \tilde E^\bullet(X) \overset{i^\ast}{\longrightarrow} \tilde E^\bullet(A) \,. </annotation></semantics></math></div></li> </ol> <p>We say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">\tilde E^\bullet</annotation></semantics></math> is <strong>additive</strong> if in addition</p> <ul> <li id="WedgeAxiom"> <p><strong>(<a class="existingWikiWord" href="/nlab/show/wedge+axiom">wedge axiom</a>)</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i\}_{i \in I} </annotation></semantics></math> any set of pointed CW-complexes, then the canonical comparison morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mo>∨</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde E^\bullet(\vee_{i \in I} X_i) \longrightarrow \prod_{i \in I} \tilde E^\bullet(X_i) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, from the functor applied to their <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a>, example <a class="maruku-ref" href="#WedgeSumAsCoproduct"></a>, to the <a class="existingWikiWord" href="/nlab/show/product">product</a> of its values on the wedge summands, .</p> </li> </ul> <p>We say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">\tilde E^\bullet</annotation></semantics></math> is <strong>ordinary</strong> if its value on the <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">S^0</annotation></semantics></math> is concentrated in degree 0:</p> <ul> <li><strong>(Dimension)</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo>≠</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>𝕊</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\tilde E^{\bullet\neq 0}(\mathbb{S}^0) \simeq 0</annotation></semantics></math>.</li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of reduced cohomology theories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo>⟶</mo><msup><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; \tilde E^\bullet \longrightarrow \tilde F^\bullet </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> between the underlying functors which is compatible with the suspension isomorphisms in that all the following <a class="existingWikiWord" href="/nlab/show/commuting+square">squares commute</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><msup><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>σ</mi> <mi>E</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>σ</mi> <mi>F</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><mi>Σ</mi><mi>X</mi></mrow></msub></mrow></mover></mtd> <mtd><msup><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde E^\bullet(X) &\overset{\eta_X}{\longrightarrow}& \tilde F^\bullet(X) \\ {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}} \\ \tilde E^{\bullet + 1}(\Sigma X) &\overset{\eta_{\Sigma X}}{\longrightarrow}& \tilde F^{\bullet + 1}(\Sigma X) } \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#AguilarGitlerPrieto02">AGP 02, def. 12.1.4</a>)</p> <p>We may rephrase this more intrinsically and more generally:</p> <div class="num_defn" id="GeneralizedCohomologyOnGeneralInfinityCategory"> <h6 id="definition_3">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> with <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pushouts">(∞,1)-pushouts</a>, and with a <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">0 \in \mathcal{C}</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒞</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>↦</mo><mn>0</mn><munder><mo>⊔</mo><mi>X</mi></munder><mn>0</mn></mrow><annotation encoding="application/x-tex">\Sigma \colon \mathcal{C} \to \mathcal{C}\colon X\mapsto 0 \underset{X}{\sqcup} 0</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>.</p> <p>A <strong>reduced generalized cohomology theory</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>⟶</mo><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex"> E^\bullet \;\colon \; Ho(\mathcal{C})^{op} \longrightarrow Ab^{\mathbb{Z}} </annotation></semantics></math></div> <p>(from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28infinity%2C1%29-category">homotopy category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a>);</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a> (“<a class="existingWikiWord" href="/nlab/show/suspension+isomorphisms">suspension isomorphisms</a>”) of degree +1</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mo>⟶</mo><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex"> \sigma \; \colon \; H^\bullet \longrightarrow H^{\bullet+1} \circ \Sigma </annotation></semantics></math></div></li> </ol> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">H^\bullet</annotation></semantics></math></p> <ol> <li> <p>takes small <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> to <a class="existingWikiWord" href="/nlab/show/products">products</a>;</p> </li> <li> <p>takes <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber+sequences">homotopy cofiber sequences</a> to <a class="existingWikiWord" href="/nlab/show/exact+sequences">exact sequences</a>.</p> </li> </ol> </div> <div class="num_defn" id="ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory"> <h6 id="definition_4">Definition</h6> <p>Given a generalized cohomology theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>H</mi> <mo>•</mo></msup><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(H^\bullet,\sigma)</annotation></semantics></math> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> as in def. <a class="maruku-ref" href="#GeneralizedCohomologyOnGeneralInfinityCategory"></a>, and given a <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber+sequence">homotopy cofiber sequence</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi><mover><mo>⟶</mo><mi>g</mi></mover><mi>Z</mi><mover><mo>⟶</mo><mrow><mi>coker</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mover><mi>Σ</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \stackrel{f}{\longrightarrow} Y \stackrel{g}{\longrightarrow} Z \stackrel{coker(g)}{\longrightarrow} \Sigma X \,, </annotation></semantics></math></div> <p>then the corresponding <strong><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></strong> is the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mover><mi>E</mi><mo>˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>σ</mi></mover><msup><mover><mi>E</mi><mo>˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>coker</mi><mo stretchy="false">(</mo><mi>g</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mover><msup><mover><mi>E</mi><mo>˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \partial \;\colon\; \widetilde E^\bullet(X) \stackrel{\sigma}{\longrightarrow} \widetilde E^{\bullet+1}(\Sigma X) \stackrel{coker(g)^\ast}{\longrightarrow} \widetilde E^{\bullet+1}(Z) \,. </annotation></semantics></math></div></div> <div class="num_prop" id="LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/connecting+homomorphisms">connecting homomorphisms</a> of def. <a class="maruku-ref" href="#ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory"></a> are part of <a class="existingWikiWord" href="/nlab/show/long+exact+sequences">long exact sequences</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mover><mo>⟶</mo><mo>∂</mo></mover><msup><mover><mi>E</mi><mo>˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mover><mi>E</mi><mo>˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mover><mi>E</mi><mo>˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>∂</mo></mover><msup><mover><mi>E</mi><mo>˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \stackrel{\partial}{\longrightarrow} \widetilde E^{\bullet}(Z) \longrightarrow \widetilde E^\bullet(Y) \longrightarrow \widetilde E^\bullet(X) \stackrel{\partial}{\longrightarrow} \widetilde E^{\bullet+1}(Z) \to \cdots \,. </annotation></semantics></math></div></div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+generalized+cohomology">long exact sequence in generalized cohomology</a></em>.</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the defining exactness of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#GeneralizedCohomologyOnGeneralInfinityCategory"></a>, and the way this appears in def. <a class="maruku-ref" href="#ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory"></a>, using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> is by definition an isomorphism.</p> </div> <h3 id="UnreducedCohomology">Unreduced cohomology</h3> <p>In the following a <em>pair</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,A)</annotation></semantics></math> refers to a <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> inclusion of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> (<a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math>. Whenever only one space is mentioned, the subspace is assumed to be the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \emptyset)</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>CW</mi> <mo>↪</mo></msubsup></mrow><annotation encoding="application/x-tex">Top_{CW}^{\hookrightarrow}</annotation></semantics></math> for the category of such pairs (the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>CW</mi></msub></mrow><annotation encoding="application/x-tex">Top_{CW}</annotation></semantics></math> on the inclusions). We identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>CW</mi></msub><mo>↪</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mo>↪</mo></msubsup></mrow><annotation encoding="application/x-tex">Top_{CW} \hookrightarrow Top_{CW}^{\hookrightarrow}</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \mapsto (X,\emptyset)</annotation></semantics></math>.</p> <div class="num_defn" id="GeneralizedCohomologyTheory"> <h6 id="definition_5">Definition</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a></em> (unreduced, <a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative</a>) is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo>:</mo><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mo>↪</mo></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex"> E^\bullet : (Top_{CW}^{\hookrightarrow})^{op} \to Ab^{\mathbb{Z}} </annotation></semantics></math></div> <p>to the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a>, as well as a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> of degree +1, to be called the <strong><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></strong>, of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \delta_{(X,A)} \;\colon\; E^\bullet(A, \emptyset) \to E^{\bullet + 1}(X, A) \,. </annotation></semantics></math></div> <p>such that:</p> <ol> <li> <p><strong>(homotopy invariance)</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \colon (X_1,A_1) \to (X_2,A_2)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> of pairs, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(f) \;\colon\; E^\bullet(X_2,A_2) \stackrel{\simeq}{\longrightarrow} E^\bullet(X_1,A_1) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>;</p> </li> <li id="ExactnessUnreduced"> <p><strong>(exactness)</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math> the induced sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>δ</mi></mover><msup><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \cdots \to E^n(X, A) \longrightarrow E^n(X) \longrightarrow E^n(A) \stackrel{\delta}{\longrightarrow} E^{n+1}(X, A) \to \cdots </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>.</p> </li> <li id="excision"> <p><strong>(<a class="existingWikiWord" href="/nlab/show/excision">excision</a>)</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↪</mo><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \hookrightarrow A \hookrightarrow X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>U</mi><mo>¯</mo></mover><mo>⊂</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\overline{U} \subset Int(A)</annotation></semantics></math>, then the natural inclusion of the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i \colon (X-U, A-U) \hookrightarrow (X, A)</annotation></semantics></math> induces an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(i) \;\colon\; E^n(X, A) \overset{\simeq}{\longrightarrow} E^n(X-U, A-U) </annotation></semantics></math></div></li> </ol> <p>We say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> is <strong>additive</strong> if it takes <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> to <a class="existingWikiWord" href="/nlab/show/products">products</a>:</p> <ul> <li> <p><strong>(additivity)</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, A) = \coprod_i (X_i, A_i)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, then the canonical comparison morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mi>i</mi></munder><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^n(X, A) \overset{\simeq}{\longrightarrow} \prod_i E^n(X_i, A_i) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> from the value on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,A)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/product">product</a> of values on the summands.</p> </li> </ul> <p>We say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> is <strong>ordinary</strong> if its value on the point is concentrated in degree 0</p> <ul> <li><strong>(Dimension)</strong>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mrow><mo>•</mo><mo>≠</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E^{\bullet \neq 0}(\ast,\emptyset) = 0</annotation></semantics></math>.</li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of unreduced cohomology theories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mo>•</mo></msup><mo>⟶</mo><msup><mi>F</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; E^\bullet \longrightarrow F^\bullet </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these <a class="existingWikiWord" href="/nlab/show/commuting+square">squares commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mover></mtd> <mtd><msup><mi>F</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>δ</mi> <mi>E</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>δ</mi> <mi>F</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mover></mtd> <mtd><msup><mi>F</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ E^\bullet(A,\emptyset) &\overset{\eta_{(A,\emptyset)}}{\longrightarrow}& F^\bullet(A,\emptyset) \\ {}^{\mathllap{\delta_E}}\downarrow && \downarrow^{\mathrlap{\delta_F}} \\ E^{\bullet +1}(X,A) &\overset{\eta_{(X,A)}}{\longrightarrow}& F^{\bullet +1}(X,A) } \,. </annotation></semantics></math></div></div> <p>e.g. (<a href="#AguilarGitlerPrieto02">AGP 02, def. 12.1.1</a>).</p> <div class="num_defn" id="AlternativeFormulationOfExcisionAxiom"> <h6 id="lemma">Lemma</h6> <p>The excision axiom in def. <a class="maruku-ref" href="#GeneralizedCohomologyTheory"></a> is equivalent to the following statement:</p> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A,B \hookrightarrow X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = Int(A) \cup Int(B)</annotation></semantics></math>, then the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i \colon (A, A \cap B) \longrightarrow (X,B) </annotation></semantics></math></div> <p>induces an isomorphism,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i^\ast \;\colon\; E^\bullet(X, B) \overset{\simeq}{\longrightarrow} E^\bullet(A, A \cap B) </annotation></semantics></math></div></div> <p>(e.g <a href="#Switzer75">Switzer 75, 7.2</a>)</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>In one direction, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> satisfies the original excision axiom. Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo lspace="0em" rspace="thinmathspace">Int</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = \Int(A) \cup Int(B)</annotation></semantics></math>, set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>≔</mo><mi>X</mi><mo>−</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">U \coloneqq X-A</annotation></semantics></math> and observe that</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><mover><mi>U</mi><mo>¯</mo></mover></mtd> <mtd><mo>=</mo><mover><mrow><mi>X</mi><mo>−</mo><mi>A</mi></mrow><mo>¯</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>X</mi><mo>−</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⊂</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \overline{U} & = \overline{X-A} \\ & = X- Int(A) \\ & \subset Int(B) \end{aligned} </annotation></semantics></math></div> <p>and that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>B</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X-U, B-U) = (A, A \cap B) \,. </annotation></semantics></math></div> <p>Hence the excision axiom implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(X, B) \overset{\simeq}{\longrightarrow} E^\bullet(A, A \cap B)</annotation></semantics></math>.</p> <p>Conversely, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> satisfies the alternative condition. Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↪</mo><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \hookrightarrow A \hookrightarrow X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>U</mi><mo>¯</mo></mover><mo>⊂</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\overline{U} \subset Int(A)</annotation></semantics></math>, observe that we have a cover</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><mi>Int</mi><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mover><mi>U</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>∪</mo><mo lspace="0em" rspace="thinmathspace">Int</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⊃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∪</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Int(X-U) \cup Int(A) & = (X - \overline{U}) \cup \Int(A) \\ & \supset (X - Int(A)) \cup Int(A) \\ & = X \end{aligned} </annotation></semantics></math></div> <p>and that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X-U, (X-U) \cap A) = (X-U, A - U) \,. </annotation></semantics></math></div> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(X-U,A-U) \simeq E^\bullet(X-U, (X-U)\cap A) \simeq E^\bullet(X,A) \,. </annotation></semantics></math></div></div> <p>The following lemma shows that the dependence in pairs of spaces in a generalized cohomology theory is really a stand-in for evaluation on <a class="existingWikiWord" href="/nlab/show/homotopy+cofibers">homotopy cofibers</a> of inclusions.</p> <div class="num_lemma" id="EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient"> <h6 id="lemma_2">Lemma</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> be an cohomology theory, def. <a class="maruku-ref" href="#GeneralizedCohomologyTheory"></a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math>. Then there is an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(X,A) \stackrel{\simeq}{\longrightarrow} E^\bullet(X \cup Cone(A), \ast) </annotation></semantics></math></div> <p>between the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> on the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,A)</annotation></semantics></math> and its value on the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> of the inclusion, relative to a basepoint.</p> <p>If moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math> is (the <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of) a <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> inclusion, then also the morphism in cohomology induced from the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \;\colon\; (X,A)\longrightarrow (X/A, \ast)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(p) \;\colon\; E^\bullet(X/A,\ast) \longrightarrow E^\bullet(X,A) \,. </annotation></semantics></math></div></div> <p>(e.g <a href="#AguilarGitlerPrieto02">AGP 02, corollary 12.1.10</a>)</p> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>≔</mo><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>−</mo><mi>A</mi><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \coloneqq (Cone(A)-A \times \{0\}) \hookrightarrow Cone(A)</annotation></semantics></math>, the cone on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> minus the base <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. We have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>−</mo><mi>U</mi><mo>,</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ( X\cup Cone(A)-U, Cone(A)-U) \simeq (X,A) </annotation></semantics></math></div> <p>and hence the first isomorphism in the statement is given by the excision axiom followed by homotopy invariance (along the contraction of the cone to the point).</p> <p>Next consider the quotient of the <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a> of the inclusion:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ( X\cup Cone(A), Cone(A) ) \longrightarrow (X/A,\ast) \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math> is a cofibration, then this is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cone(A)</annotation></semantics></math> is contractible and since by the dual <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X \cup Cone(A)\to X/A</annotation></semantics></math> is a weak homotopy equivalence, hence a homotopy equivalence on CW-complexes.</p> <p>Hence now we get a composite isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(X/A,\ast) \overset{\simeq}{\longrightarrow} E^\bullet( X\cup Cone(A), Cone(A) ) \overset{\simeq}{\longrightarrow} E^\bullet(X,A) \,. </annotation></semantics></math></div></div> <div class="num_example" id="GeneralizedCohomologyOnHomotopyQuotientMaps"> <h6 id="example_2">Example</h6> <p>As an important special case of : Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p\colon (Cone(X), X) \to (\Sigma X,\{x\})</annotation></semantics></math> the quotient map from the reduced cone on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(p) \;\colon\; E^\bullet(Cone(X),X) \overset{\simeq}{\longrightarrow} E^\bullet(\Sigma X, \{x\}) </annotation></semantics></math></div> <p>is an isomorphism.</p> </div> <div class="num_prop" id="ExactSequenceOfATriple"> <h6 id="proposition_2">Proposition</h6> <p><strong>(exact sequence of a triple)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> an unreduced generalized cohomology theory, def. <a class="maruku-ref" href="#GeneralizedCohomologyTheory"></a>, then every inclusion of two consecutive subspaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>↪</mo><mi>Y</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> Z \hookrightarrow Y \hookrightarrow X </annotation></semantics></math></div> <p>induces a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of cohomology groups 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>⋯</mi><mo>→</mo><msup><mi>E</mi> <mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover></mover><msup><mi>E</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msup><mi>E</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msup><mi>E</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \cdots \to E^{q-1}(Y,Z) \stackrel{\bar \delta}{\longrightarrow} E^q(X,Y) \stackrel{}{\longrightarrow} E^q(X,Z) \stackrel{}{\longrightarrow} E^q(Y,Z) \to \cdots </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>δ</mi></mover><msup><mi>E</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \bar \delta \;\colon \; E^{q-1}(Y,Z) \longrightarrow E^{q-1}(Y) \stackrel{\delta}{\longrightarrow} E^{q}(X,Y) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Apply the <a class="existingWikiWord" href="/nlab/show/braid+lemma">braid lemma</a> to the interlocking long exact sequences of the three pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Z)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,Z)</annotation></semantics></math>. See <a href="braid+lemma#ExactSequenceForTripleInGeneralizedHomology">here</a> for details.</p> <p>The dual braid diagram for <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a> is this:</p> <p><img src="http://www.ncatlab.org/nlab/files/BraidDiagramForHomologyOnTripled.jpg" width="500" /></p> <p>(graphics from <a href="http://math.stackexchange.com/a/1180681/58526">this Maths.SE comment</a>)</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The exact sequence of a triple in prop. <a class="maruku-ref" href="#ExactSequenceOfATriple"></a> is what gives rise to the <a class="existingWikiWord" href="/nlab/show/Cartan-Eilenberg+spectral+sequence">Cartan-Eilenberg spectral sequence</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-cohomology of a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_example" id="ExtractingSuspensionIsomorphismFromUnreducedCohomology"> <h6 id="example_3">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mo stretchy="false">(</mo><msub><mi>I</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Cone(X) = (X \wedge (I_+))/X</annotation></semantics></math> its reduced <a class="existingWikiWord" href="/nlab/show/cone">cone</a>, the long exact sequence of the triple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo>,</mo><mi>X</mi><mo>,</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\{x\}, X, Cone(X))</annotation></semantics></math>, prop. <a class="maruku-ref" href="#ExactSequenceOfATriple"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≃</mo><msup><mi>E</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover></mover><msup><mi>E</mi> <mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \simeq E^q(Cone(X), \{x\}) \longrightarrow E^q(X,\{x\}) \overset{\bar \delta}{\longrightarrow} E^{q+1}(Cone(X),X) \longrightarrow E^{q+1}(Cone(X), \{x\}) \simeq 0 </annotation></semantics></math></div> <p>exhibits the <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar \delta</annotation></semantics></math> here as an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>E</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \bar \delta \;\colon\; E^q(X,\{x\}) \overset{\simeq}{\longrightarrow} E^{q+1}(Cone(X),X) \,. </annotation></semantics></math></div> <p>This is the <em><a class="existingWikiWord" href="/nlab/show/suspension+isomorphism">suspension isomorphism</a></em> extracted from the unreduced cohomology theory, see def. <a class="maruku-ref" href="#FromUnreducedToReducedCohomology"></a> below.</p> </div> <div class="num_prop" id="MayerVietorisSequenceInGeneralizedCohomology"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Mayer-Vietoris+sequence">Mayer-Vietoris sequence</a>)</strong></p> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> an unreduced cohomology theory, def. <a class="maruku-ref" href="#GeneralizedCohomologyTheory"></a>. Given a topological space covered by the <a class="existingWikiWord" href="/nlab/show/interior">interior</a> of two spaces as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = Int(A) \cup Int(B)</annotation></semantics></math>, then for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>A</mi><mo>∩</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">C \subset A \cap B</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of cohomology groups 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>⋯</mi><mo>→</mo><msup><mi>E</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover></mover><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>E</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \to E^{n-1}(A \cap B , C) \overset{\bar \delta}{\longrightarrow} E^n(X,C) \longrightarrow E^n(A,C) \oplus E^n(B,C) \longrightarrow E^n(A \cap B, C) \to \cdots \,. </annotation></semantics></math></div></div> <p>e.g. (<a href="#Switzer75">Switzer 75, theorem 7.19</a>, <a href="#AguilarGitlerPrieto02">Aguilar-Gitler-Prieto 02, theorem 12.1.22</a>, see also at <em><a class="existingWikiWord" href="/nlab/show/Brown-Gersten+property">Brown-Gersten property</a></em>)</p> <h2 id="RelationBetweenReducedAndUnreduced">Relation between reduced and unreduced cohomology</h2> <div class="num_defn" id="FromUnreducedToReducedCohomology"> <h6 id="definition_6">Definition</h6> <p><strong>(unreduced to reduced cohomology)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> be an unreduced cohomology theory, def. <a class="maruku-ref" href="#GeneralizedCohomologyTheory"></a>. Define a reduced cohomology theory, def. <a class="maruku-ref" href="#ReducedGeneralizedCohomology"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\tilde E^\bullet, \sigma)</annotation></semantics></math> as follows.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \colon \ast \to X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a>, set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde E^\bullet(X,x) \coloneqq E^\bullet(X,\{x\}) \,. </annotation></semantics></math></div> <p>This is clearly <a class="existingWikiWord" href="/nlab/show/functor">functorial</a>. Take the <a class="existingWikiWord" href="/nlab/show/suspension+isomorphism">suspension isomorphism</a> to be the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></mover><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \sigma \;\colon\; \tilde E^{\bullet+1}(\Sigma X) = E^{\bullet+1}(\Sigma X, \{x\}) \overset{E^\bullet(p)}{\longrightarrow} E^{\bullet+1}(Cone(X),X) \overset{\bar \delta^{-1}}{\longrightarrow} E^\bullet(X,\{x\}) = \tilde E^{\bullet}(X) </annotation></semantics></math></div> <p>of the isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^\bullet(p)</annotation></semantics></math> from example <a class="maruku-ref" href="#GeneralizedCohomologyOnHomotopyQuotientMaps"></a> and the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> of the isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\bar \delta</annotation></semantics></math> from example <a class="maruku-ref" href="#ExtractingSuspensionIsomorphismFromUnreducedCohomology"></a>.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>The construction in def. <a class="maruku-ref" href="#FromUnreducedToReducedCohomology"></a> indeed gives a reduced cohomology theory.</p> </div> <p>(e.g. <a href="#Switzer75">Switzer 75, 7.34</a>)</p> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>We need to check the exactness axiom given any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A\hookrightarrow X</annotation></semantics></math>. By lemma <a class="maruku-ref" href="#EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient"></a> we have an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">{</mo><mo>*</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde E^\bullet(X \cup Cone(A)) = E^\bullet(X \cup Cone(A), \{\ast\}) \overset{\simeq}{\longrightarrow} E^\bullet(X,A) \,. </annotation></semantics></math></div> <p>Unwinding the constructions shows that this makes the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mo stretchy="false">{</mo><mi>a</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde E^\bullet(X\cup Cone(A)) &\overset{\simeq}{\longrightarrow}& E^\bullet(X,A) \\ \downarrow && \downarrow \\ \tilde E^\bullet(X) &=& E^\bullet(X,\{x\}) \\ \downarrow && \downarrow \\ \tilde E^\bullet(A) &=& E^\bullet(A,\{a\}) } \,, </annotation></semantics></math></div> <p>where the vertical sequence on the right is exact by prop. <a class="maruku-ref" href="#ExactSequenceOfATriple"></a>. Hence the left vertical sequence is exact.</p> </div> <div class="num_defn" id="ReducedToUnreducedGeneralizedCohomology"> <h6 id="definition_7">Definition</h6> <p><strong>(reduced to unreduced cohomology)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\tilde E^\bullet, \sigma)</annotation></semantics></math> be a reduced cohomology theory, def. <a class="maruku-ref" href="#ReducedGeneralizedCohomology"></a>. Define an unreduced cohomolog theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math>, def. <a class="maruku-ref" href="#GeneralizedCohomologyTheory"></a>, by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>+</mo></msub><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(X,A) \coloneqq \tilde E^\bullet( X_+ \cup Cone(A_+)) </annotation></semantics></math></div> <p>and let the connecting homomorphism be as in def. <a class="maruku-ref" href="#ConnectinHomomorphismForCohomologyTheoryOnInfinityCategory"></a>.</p> </div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>The construction in def. <a class="maruku-ref" href="#ReducedToUnreducedGeneralizedCohomology"></a> indeed yields an unreduced cohomology theory.</p> </div> <p>e.g. (<a href="#Switzer75">Switzer 75, 7.35</a>)</p> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>Exactness holds by prop. <a class="maruku-ref" href="#LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory"></a>. For excision, it is sufficient to consider the alternative formulation of lemma <a class="maruku-ref" href="#AlternativeFormulationOfExcisionAxiom"></a>. For CW-inclusions, this follows immediately with lemma <a class="maruku-ref" href="#EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient"></a>.</p> </div> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>The constructions of def. <a class="maruku-ref" href="#ReducedToUnreducedGeneralizedCohomology"></a> and def. <a class="maruku-ref" href="#FromUnreducedToReducedCohomology"></a> constitute a pair of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> between then <a class="existingWikiWord" href="/nlab/show/categories">categories</a> of reduced cohomology theories, def. <a class="maruku-ref" href="#ReducedGeneralizedCohomology"></a> and unreduced cohomology theories, def. <a class="maruku-ref" href="#GeneralizedCohomologyTheory"></a> which exhbit an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>.</p> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>(…careful with checking the respect for suspension iso and connecting homomorphism..)</p> <p>To see that there are <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a> relating the two composites of these two functors to the identity:</p> <p>One composite is</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><msup><mi>E</mi> <mo>•</mo></msup></mtd> <mtd><mo>↦</mo><mo stretchy="false">(</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>↦</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↦</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>E</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mo>•</mo></msup><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>↦</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>+</mo></msub><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} E^\bullet & \mapsto (\tilde E^\bullet \colon (X,x) \mapsto E^\bullet(X,\{x\})) \\ & \mapsto ((E')^\bullet \colon (X,A) \mapsto E^\bullet( X_+ \cup Cone(A_+) ), \ast) \end{aligned} \,, </annotation></semantics></math></div> <p>where on the right we have, from the construction, the reduced mapping cone of the original inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math> with a base point adjoined. That however is isomorphic to the unreduced mapping cone of the original inclusion. With this the natural isomorphism is given by lemma <a class="maruku-ref" href="#EvaluationOfCohomologyTheoryOnGoodPairIsEvaluationOnQuotient"></a>.</p> <p>The other composite is</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><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup></mtd> <mtd><mo>↦</mo><mo stretchy="false">(</mo><msup><mi>E</mi> <mo>•</mo></msup><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>↦</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>+</mo></msub><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↦</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mover><mi>E</mi><mo stretchy="false">˜</mo></mover><mo>′</mo><msup><mo stretchy="false">)</mo> <mo>•</mo></msup><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>↦</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>+</mo></msub><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><msub><mo>*</mo> <mo>+</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \tilde E^\bullet & \mapsto (E^\bullet \colon (X,A) \mapsto \tilde E^\bullet(X_+ \cup Cone(A_+))) \\ & \mapsto ((\tilde E')^\bullet \colon X \mapsto \tilde E^\bullet(X_+ \cup Cone(*_+))) \end{aligned} </annotation></semantics></math></div> <p>where on the right we have the reduced mapping cone of the point inclusion with a point adoined. As before, this is isomorphic to the unreduced mapping cone of the point inclusion. That finally is clearly homotopy equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and so now the natural isomorphism follows with homotopy invariance.</p> </div> <p>Finally we record the following basic relation between reduced and unreduced cohomology:</p> <div class="num_prop" id="UnreducedCohomologyIsReducedPlusPointValue"> <h6 id="proposition_6">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> be an unreduced cohomology theory, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">\tilde E^\bullet</annotation></semantics></math> its reduced cohomology theory from def. <a class="maruku-ref" href="#FromUnreducedToReducedCohomology"></a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\ast)</annotation></semantics></math> a pointed topological space, then there is an identification</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(X) \simeq \tilde E^\bullet(X) \oplus E^\bullet(\ast) </annotation></semantics></math></div> <p>of the unreduced cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of the reduced cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and the unreduced cohomology of the base point.</p> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>The pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\ast \hookrightarrow X</annotation></semantics></math> induces the sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msup><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>δ</mi></mover><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>δ</mi></mover><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \cdots \to E^{\bullet-1}(\ast) \stackrel{\delta}{\longrightarrow} \tilde E^\bullet(X) \stackrel{}{\longrightarrow} E^\bullet(X) \stackrel{}{\longrightarrow} E^\bullet(\ast) \stackrel{\delta}{\longrightarrow} \tilde E^{\bullet+1}(X) \to \cdots </annotation></semantics></math></div> <p>which by the exactness clause in def. <a class="maruku-ref" href="#GeneralizedCohomologyTheory"></a> is <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact</a>.</p> <p>Now since the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast \to X \to \ast</annotation></semantics></math> is the identity, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^\bullet(X) \to E^\bullet(\ast)</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/section">section</a> and so is in particular an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>. Therefore, by exactness, the <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a> vanishes, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\delta = 0</annotation></semantics></math> and we have a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to \tilde E^\bullet(X) \stackrel{}{\longrightarrow} E^\bullet(X) \stackrel{}{\longrightarrow} E^\bullet(\ast) \to 0 </annotation></semantics></math></div> <p>with the right map an epimorphism. Hence this is a <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a> and the statement follows.</p> </div> <h2 id="brown_functoriality">Brown functoriality</h2> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>Given a generalized cohomology functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex">E^\bullet \colon Ho(\mathcal{C})^{op}\to Ab^{\mathbb{Z}}</annotation></semantics></math>, def. <a class="maruku-ref" href="#GeneralizedCohomologyOnGeneralInfinityCategory"></a>, its underlying <a class="existingWikiWord" href="/nlab/show/Set">Set</a>-valued functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mi>Ab</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">H^n \colon Ho(\mathcal{C})^{op}\to Ab\to Set</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/Brown+functors">Brown functors</a>, def. <a class="maruku-ref" href="#BrownFunctorOnInfinityCategory"></a>.</p> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>The first condition on a <a class="existingWikiWord" href="/nlab/show/Brown+functor">Brown functor</a> holds by definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">H^\bullet</annotation></semantics></math>. For the second condition, given a homotopy pushout square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msub><mi>Y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X_1 &\stackrel{f_1}{\longrightarrow}& Y_1 \\ \downarrow^{} && \downarrow \\ X_2 &\stackrel{f_2}{\longrightarrow}& Y_2 } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, consider the induced morphism of the <a class="existingWikiWord" href="/nlab/show/long+exact+sequences">long exact sequences</a> given by prop. <a class="maruku-ref" href="#LongExactSequenceOfACohomologyTheoryOnAnInfinityCategory"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>Y</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>f</mi> <mn>2</mn> <mo>*</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>Y</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>f</mi> <mn>1</mn> <mo>*</mo></msubsup></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mi>coker</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ H^\bullet(coker(f_2)) &\longrightarrow& H^\bullet(Y_2) &\stackrel{f^\ast_2}{\longrightarrow}& H^\bullet(X_2) &\stackrel{}{\longrightarrow}& H^{\bullet+1}(\Sigma coker(f_2)) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow && \downarrow && \downarrow^{\mathrlap{\simeq}} \\ H^\bullet(coker(f_1)) &\longrightarrow& H^\bullet(Y_1) &\stackrel{f^\ast_1}{\longrightarrow}& H^\bullet(X_1) &\stackrel{}{\longrightarrow}& H^{\bullet+1}(\Sigma coker(f_1)) } </annotation></semantics></math></div> <p>Here the outer vertical morphisms are <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, as shown, due to the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> (see also at <em><a href="homotopy+pullback#FiberwiseRecognitionInStableCase">fiberwise recognition of stable homotopy pushouts</a></em>).</p> <p>This means that the <a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a> applies to this diagram. Inspection shows that this implies the claim.</p> </div> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>: <a class="existingWikiWord" href="/nlab/show/Eilenberg-Mac+Lane+spectrum">Eilenberg-Mac Lane spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a>: <a class="existingWikiWord" href="/nlab/show/K-theory+spectrum">K-theory spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <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/Morava+K-theory">Morava K-theory</a>, <a class="existingWikiWord" href="/nlab/show/integral+Morava+K-theory">integral Morava K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a></p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="expression_by_ordinary_cohomology_via_atiyahhirzebruch_spectral_sequence">Expression by ordinary cohomology via Atiyah-Hirzebruch spectral sequence</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a> serves to express generalized cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> in terms of ordinary cohomology with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E^\bullet(\ast)</annotation></semantics></math>.</p> <h3 id="WhiteheadTheorem">Whitehead theorem</h3> <div class="num_prop"> <h6 id="proposition_8">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>⟶</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\phi \colon E \longrightarrow F</annotation></semantics></math> be a morphism of reduced generalized (co-)homology functors, def. <a class="maruku-ref" href="#ReducedGeneralizedCohomology"></a> (a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>) such that its component</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi(S^0) \colon E(S^0) \longrightarrow F(S^0) </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(X)\colon E(X)\to F(X)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> with a <a class="existingWikiWord" href="/nlab/show/finite+number">finite number</a> of cells. If both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> satisfy the <a class="existingWikiWord" href="/nlab/show/wedge+axiom">wedge axiom</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(X)</annotation></semantics></math> is an isomorphism for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>, not necessarily finite.</p> </div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>/<a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> functors a proof of this is in (<a href="#EilenbergSteenrod52">Eilenberg-Steenrod 52, section III.10</a>). From this the general statement follows (e.g. <a href="#Kochman96">Kochman 96, theorem 3.4.3, corollary 4.2.8</a>) via the <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> of the <a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a> (the classical result gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> induces an isomorphism between the second pages of the AHSSs for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>). A complete proof of the general result is also given as (<a href="#Switzer75">Switzer 75, theorem 7.55, theorem 7.67</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <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/generalized+homology+theory">generalized homology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kronecker+pairing">Kronecker pairing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bivariant+cohomology+theory">bivariant cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Dold+character">Chern-Dold character</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> is conjecturally <a class="existingWikiWord" href="/nlab/show/categorical+semantics">∞-categorical</a> <a class="existingWikiWord" href="/nlab/show/semantics">semantics</a> of <a class="existingWikiWord" href="/nlab/show/linear+homotopy+type+theory">linear homotopy type theory</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/linear+homotopy+type+theory">linear homotopy type theory</a></th><th><a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/quantum+theory">quantum theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+type">linear type</a></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/module+spectrum">module</a>-)<a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/multiplicative+conjunction">multiplicative conjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/composite+system">composite system</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+linear+type">dependent linear type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-module+bundle">module spectrum bundle</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/six+operation+yoga">six operation yoga</a> in <a class="existingWikiWord" href="/nlab/show/Wirthm%C3%BCller+context">Wirthmüller context</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dual+object">dual type</a> (linear negation)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/invertible+type">invertible type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+generalized+cohomology">twist</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> (“<a class="existingWikiWord" href="/nlab/show/bra-ket">bra</a>”)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dual+type">dual</a> of <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> (“<a class="existingWikiWord" href="/nlab/show/bra-ket">ket</a>”)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bivariant+cohomology">bivariant cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantum+operators">quantum operators</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/exponential+modality">exponential modality</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> over <a class="existingWikiWord" href="/nlab/show/finite+homotopy+type">finite homotopy type</a> (of twist)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a> (<a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a>)</td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable</a> <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> over <a class="existingWikiWord" href="/nlab/show/finite+homotopy+type">finite homotopy type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah+duality">Atiyah duality</a> between <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> and <a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">(twisted) <a class="existingWikiWord" href="/nlab/show/self-dual+object">self-dual type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> coinciding with <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ambidextrous+adjunction">ambidexterity</a>, <a class="existingWikiWord" href="/nlab/show/semiadditive+%28%E2%88%9E%2C1%29-category">semiadditivity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> coinciding with <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> up to <a class="existingWikiWord" href="/nlab/show/invertible+type">invertible type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Wirthm%C3%BCller+isomorphism">Wirthmüller isomorphism</a></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 stretchy="false">(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\sum_f \dashv f^\ast)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a></td><td style="text-align: left;">pushforward in <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">(twisted-)<a class="existingWikiWord" href="/nlab/show/self-dual+object">self-duality</a>-<a href="self-dual+object#RelationToDaggerCompactStructure">induced dagger</a> of this counit</td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/twisted+Umkehr+map">twisted</a>-)<a class="existingWikiWord" href="/nlab/show/Umkehr+map">Umkehr map</a>/<a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+polynomial+functor">linear polynomial functor</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/correspondence">correspondence</a></td><td style="text-align: left;">space of <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+polynomial+functor">linear polynomial functor</a> with <a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+kernel">integral kernel</a> (<a class="existingWikiWord" href="/nlab/show/pure+motive">pure motive</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantized+Lagrangian+correspondence">prequantized Lagrangian correspondence</a>/<a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></td></tr> <tr><td style="text-align: left;">composite of this <a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a> with daggered-counit followed by unit</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+transform">integral transform</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic</a>/cohomological <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trace">trace</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>The original article on the Eilenberg–Steenrod axioms:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Norman+E.+Steenrod">Norman E. Steenrod</a>, <em>Axiomatic Approach to Homology Theory</em>, Proceedings of the National Academy of Sciences <strong>31</strong> 4 (1945) 117–120 [<a href="http://dx.doi.org/10.1073/pnas.31.4.117">doi:10.1073/pnas.31.4.117</a>]</li> </ul> <p>Further development and an expository account:</p> <ul> <li id="EilenbergSteenrod52"><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>, <em>Foundations of algebraic topology</em>, Princeton 1952 (<a href="http://www.maths.ed.ac.uk/~aar/papers/eilestee.pdf">pdf</a>, <a href="https://press.princeton.edu/books/hardcover/9780691653297/foundations-of-algebraic-topology">ISBN:9780691653297</a>)</li> </ul> <p>The concept of generalized homology obtained by discarding the dimension axiom and the observation that every <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> induces an example is due to</p> <ul> <li id="Whitehead62"><a class="existingWikiWord" href="/nlab/show/George+Whitehead">George Whitehead</a>, <em>Generalized homology theories</em>, Transactions of the American Mathematical Society, 102 (1962) 227-283 (<a href="http://www.ams.org/journals/tran/1962-102-02/S0002-9947-1962-0137117-6/S0002-9947-1962-0137117-6.pdf">pdf</a>, <a href="https://www.jstor.org/stable/1993676">jstor:1993676</a>)</li> </ul> <p>The proof that every generalized (co)homology theory arises this way (<a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a>) is due to</p> <ul> <li id="Brown62"> <p><a class="existingWikiWord" href="/nlab/show/Edgar+Brown">Edgar Brown</a>, <em>Cohomology theories</em>, Annals of Mathematics, Second Series 75: 467–484 (1962) (<a href="https://www.jstor.org/stable/1970209">jstor:1970209</a>)</p> </li> <li id="Brown65"> <p><a class="existingWikiWord" href="/nlab/show/Edgar+Brown">Edgar Brown</a>, <em>Abstract homotopy theory</em>, Trans. AMS 119 no. 1 (1965) (<a href="https://doi.org/10.1090/S0002-9947-1965-0182970-6">doi:10.1090/S0002-9947-1965-0182970-6</a>)</p> </li> </ul> <p>An early lecture note account is in</p> <ul> <li id="Adams74"><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, part III, sections 2 and 6 of <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Switzer75"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Switzer">Robert Switzer</a>, chapter 7 (and 8-12) of <em>Algebraic Topology - Homotopy and Homology</em>, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Michael+Boardman">John Michael Boardman</a>, Section 3 of: <em>Stable Operations in Generalized Cohomology</em> [<a href="https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Boardman-StableOperations.pdf" title="pdf">pdf</a>] in: <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a> (ed.) <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em> Oxford 1995 (<a href="https://doi.org/10.1016/B978-0-444-81779-2.X5000-7">doi:10.1016/B978-0-444-81779-2.X5000-7</a>)</p> </li> <li id="Kochman96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</a>, section 3.4 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</p> </li> <li id="May99"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a> chapter 18 of <em>A Concise Course on Algebraic Topology</em>, Chicago Lecture Notes 1999 (<a href="http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf">pdf</a>)</p> </li> <li id="AguilarGitlerPrieto02"> <p>Marcelo Aguilar, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, Carlos Prieto, section 12.1 of <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href="http://tocs.ulb.tu-darmstadt.de/106999419.pdf">toc pdf</a>)</p> </li> <li id="KonoTamaki02"> <p><a class="existingWikiWord" href="/nlab/show/Akira+Kono">Akira Kono</a>, <a class="existingWikiWord" href="/nlab/show/Dai+Tamaki">Dai Tamaki</a>, <em>Generalized cohomology</em>, AMS 2002, esp. chapter 2 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a class="existingWikiWord" href="/nlab/files/GeneralizedCohomology.pdf" title="pdf">pdf</a>, <a href="https://bookstore.ams.org/mmono-230">ISBN: 978-0-8218-3514-2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, chapter II, section 6 of <em><a class="existingWikiWord" href="/nlab/show/Symmetric+spectra">Symmetric spectra</a></em>, 2012 (<a href="http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf">pdf</a>)</p> </li> </ul> <p>Discussion in the further generality of <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> is in</p> <ul> <li id="tomDieck79"><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, section 7 of <em><a class="existingWikiWord" href="/nlab/show/Transformation+Groups+and+Representation+Theory">Transformation Groups and Representation Theory</a></em>, Lecture Notes in Mathematics 766, Springer 1979</li> </ul> <p>A pedagogical introduction to <a class="existingWikiWord" href="/nlab/show/spectrum">spectra</a> and generalized (Eilenberg-Steenrod) cohomology is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a href="http://math.ucr.edu/home/baez/twf_ascii/week149">TWF 149</a></li> </ul> <p>Formulation in <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+theory">(infinity,1)-category theory</a> is in</p> <ul> <li id="LurieHigherAlgebra"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 1.4.1 of <em><a class="existingWikiWord" href="/nlab/show/Higher+Algebra">Higher Algebra</a></em></li> </ul> <p>More references relating to the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a> on cohomology include:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mike+Hopkins">Mike Hopkins</a>, <em><a class="existingWikiWord" href="/nlab/show/Complex+oriented+cohomology+theories+and+the+language+of+stacks">Complex oriented cohomology theories and the language of stacks</a></em> course notes (<a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/coctalos.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/A+Survey+of+Elliptic+Cohomology+-+cohomology+theories">A Survey of Elliptic Cohomology - cohomology theories</a></em></p> </li> </ul> <p>Formulation in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> (cf. <em><a class="existingWikiWord" href="/nlab/show/cohomology+in+homotopy+type+theory">cohomology in homotopy type theory</a></em>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Evan+Cavallo">Evan Cavallo</a>, Section 3.2 of: <em>Synthetic Cohomology in Homotopy Type Theory</em> (2015) [<a href="https://staff.math.su.se/evan.cavallo/works/thesis15.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Cavallo-CohomologyInHoTT.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Floris+van+Doorn">Floris van Doorn</a>, around Def. 5.4.2 in: <em>On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory</em> (2018) [<a href="https://arxiv.org/abs/1808.10690">arXiv:1808.10690</a>, <a href="http://florisvandoorn.com/papers/dissertation.pdf">pdf</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 17, 2024 at 01:37:05. See the <a href="/nlab/history/generalized+%28Eilenberg-Steenrod%29+cohomology" 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/generalized+%28Eilenberg-Steenrod%29+cohomology" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/6930/#Item_9">Discuss</a><span class="backintime"><a href="/nlab/revision/generalized+%28Eilenberg-Steenrod%29+cohomology/90" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/generalized+%28Eilenberg-Steenrod%29+cohomology" accesskey="S" class="navlink" id="history" rel="nofollow">History (90 revisions)</a> <a href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology/cite" style="color: black">Cite</a> <a href="/nlab/print/generalized+%28Eilenberg-Steenrod%29+cohomology" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/generalized+%28Eilenberg-Steenrod%29+cohomology" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>