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> Wu class 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> Wu class </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/3475/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_stiefelwhitney_classes'>Relation to Stiefel-Whitney classes</a></li> <li><a href='#RelationToPontryaginClasses'>Relation to Pontryagin classes</a></li> </ul> <li><a href='#applications'>Applications</a></li> <ul> <li><a href='#to_higher_dimensional_chernsimons_theory'>To higher dimensional Chern-Simons theory</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><em>Wu classes</em> are a type of <a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> that refine the <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+classes">Stiefel-Whitney classes</a>.</p> <h2 id="definition">Definition</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> equipped with a class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E : X \to B SO(n)</annotation></semantics></math> (a real <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> of some <a class="existingWikiWord" href="/nlab/show/rank">rank</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>), write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mi>k</mi></msub><mo>∈</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> w_k \in H^k(X, \mathbb{Z}_2) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+classes">Stiefel-Whitney classes</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Moreover, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∪</mo><mo>:</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>×</mo><msup><mi>H</mi> <mi>l</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>k</mi><mo>+</mo><mi>l</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \cup : H^k(X, \mathbb{Z}_2) \times H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a> and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msup><mi>H</mi> <mi>l</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>k</mi><mo>+</mo><mi>l</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sq^k(-) : H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/Steenrod+square">Steenrod square</a> operations.</p> <div class="num_defn" id="WuClassesBySteenrodSquares"> <h6 id="definition_2">Definition</h6> <p>The <strong>Wu class</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ν</mi> <mi>k</mi></msub><mo>∈</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \nu_k \in H^k(X,\mathbb{Z}_2) </annotation></semantics></math></div> <p>is defined to be the class that “represents” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sq^k(-)</annotation></semantics></math> under the cup product, in the sense that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in H^{n-k}(X, \mathbb{Z}_2)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is the dimension 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>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ν</mi> <mi>k</mi></msub><mo>∪</mo><mi>x</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sq^k(x) = \nu_k \cup x \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#MilnorStasheff74">Milnor-Stasheff 74, p. 131-133</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In other words this says that the lifts of Wu classes to <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> (<a class="existingWikiWord" href="/nlab/show/integral+Wu+structures">integral Wu structures</a>) are <em><a class="existingWikiWord" href="/nlab/show/characteristic+element+of+a+bilinear+form">characteristic elements</a></em> of the <a class="existingWikiWord" href="/nlab/show/intersection+product">intersection product</a> on integral cohomology, inducing <a class="existingWikiWord" href="/nlab/show/quadratic+refinements">quadratic refinements</a>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="relation_to_stiefelwhitney_classes">Relation to Stiefel-Whitney classes</h3> <p>The total <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+class">Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> is the total <a class="existingWikiWord" href="/nlab/show/Steenrod+square">Steenrod square</a> of the total Wu class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">\nu</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>w</mi><mo>=</mo><mi>Sq</mi><mo stretchy="false">(</mo><mi>ν</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> w = Sq(\nu) \,. </annotation></semantics></math></div> <p>Solving this for the components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ν</mi></mrow><annotation encoding="application/x-tex">\nu</annotation></semantics></math> in terms of the components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math>, one finds the first few Wu classes as <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a> in the Stiefel-Whitney classes as follows</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ν</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>w</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\nu_1 = w_1</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ν</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>w</mi> <mn>2</mn></msub><mo>+</mo><msubsup><mi>w</mi> <mn>1</mn> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\nu_2 = w_2 + w_1^2</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ν</mi> <mn>3</mn></msub><mo>=</mo><msub><mi>w</mi> <mn>1</mn></msub><msub><mi>w</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\nu_3 = w_1 w_2</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ν</mi> <mn>4</mn></msub><mo>=</mo><msub><mi>w</mi> <mn>4</mn></msub><mo>+</mo><msub><mi>w</mi> <mn>3</mn></msub><msub><mi>w</mi> <mn>1</mn></msub><mo>+</mo><msubsup><mi>w</mi> <mn>2</mn> <mn>2</mn></msubsup><mo>+</mo><msubsup><mi>w</mi> <mn>1</mn> <mn>4</mn></msubsup></mrow><annotation encoding="application/x-tex">\nu_4 = w_4 + w_3 w_1 + w_2^2 + w_1^4</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ν</mi> <mn>5</mn></msub><mo>=</mo><msub><mi>w</mi> <mn>4</mn></msub><msub><mi>w</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>w</mi> <mn>3</mn></msub><msubsup><mi>w</mi> <mn>1</mn> <mn>2</mn></msubsup><mo>+</mo><msubsup><mi>w</mi> <mn>2</mn> <mn>2</mn></msubsup><msub><mi>w</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>w</mi> <mn>2</mn></msub><msubsup><mi>w</mi> <mn>1</mn> <mn>3</mn></msubsup></mrow><annotation encoding="application/x-tex">\nu_5 = w_4 w_1 + w_3 w_1^2 + w_2^2 w_1 + w_2 w_1^3</annotation></semantics></math></p> </li> </ul> <p>…</p> <h3 id="RelationToPontryaginClasses">Relation to Pontryagin classes</h3> <div class="num_prop" id="InTermsOfPontryagin"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T X : X \to B SO(n)</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>T</mi><mo stretchy="false">^</mo></mover><mi>X</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat T X : X \to B Spin(n)</annotation></semantics></math>. Then the following classes in <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, built from <a class="existingWikiWord" href="/nlab/show/Pontryagin+classes">Pontryagin classes</a>, coincide with Wu-classes under mod-2-reduction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo>→</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z} \to \mathbb{Z}_2</annotation></semantics></math>:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ν</mi> <mn>4</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\nu_4 = \frac{1}{2} p_1</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ν</mi> <mn>8</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo stretchy="false">(</mo><mn>11</mn><msubsup><mi>p</mi> <mn>1</mn> <mn>2</mn></msubsup><mo>−</mo><mn>20</mn><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nu_8 = \frac{1}{8}(11 p_1^2 - 20 p_2)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ν</mi> <mn>12</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>16</mn></mfrac><mo stretchy="false">(</mo><mn>37</mn><msubsup><mi>p</mi> <mn>1</mn> <mn>3</mn></msubsup><mo>−</mo><mn>100</mn><msub><mi>p</mi> <mn>1</mn></msub><msub><mi>p</mi> <mn>2</mn></msub><mo>+</mo><mn>80</mn><msub><mi>p</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nu_{12} = \frac{1}{16}(37 p_1^3 - 100 p_1 p_2 + 80 p_3)</annotation></semantics></math>.</p> </li> </ul> <p>(all products are <a class="existingWikiWord" href="/nlab/show/cup+product">cup products</a>).</p> </div> <p>This is discussed in (<a href="#HopkinsSinger">Hopkins-Singer, page 101</a>).</p> <div class="num_cor" id="DivisibilityOfCupSquare"> <h6 id="corollary">Corollary</h6> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is 8 dimensional. Then, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in H^4(X, \mathbb{Z})</annotation></semantics></math> any integral 4-class, the expression</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∪</mo><mi>G</mi><mo>−</mo><mi>G</mi><mo>∪</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo>∈</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G \cup G - G \cup \frac{1}{2}p_1 \in H^4(X, \mathbb{Z}) </annotation></semantics></math></div> <p>is always even (divisible by 2).</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By the basic properties of Steenrod squares, we have for the 4-class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∪</mo><mi>G</mi><mo>=</mo><msup><mi>Sq</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G \cup G = Sq^4(G) \,. </annotation></semantics></math></div> <p>By the definition <a class="maruku-ref" href="#WuClassesBySteenrodSquares"></a> of Wu classes, the image of this integral class in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-coefficients equals the cup product with the Wu class</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∪</mo><mi>G</mi><mo>−</mo><mi>G</mi><mo>∪</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo>=</mo><msup><mi>Sq</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>−</mo><mi>G</mi><mo>∪</mo><msub><mi>ν</mi> <mn>4</mn></msub><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mi>mod</mi><mspace width="thickmathspace"></mspace><mn>2</mn><mo>.</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> G \cup G - G \cup \frac{1}{2}p_1 = Sq^4(G) - G \cup \nu_4 = 0 \; mod \; 2. \,, </annotation></semantics></math></div> <p>where the first step is by prop. <a class="maruku-ref" href="#InTermsOfPontryagin"></a>.</p> </div> <h2 id="applications">Applications</h2> <h3 id="to_higher_dimensional_chernsimons_theory">To higher dimensional Chern-Simons theory</h3> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The relation <a class="maruku-ref" href="#DivisibilityOfCupSquare"></a> plays a central role in the definition of the <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">7-dimensional Chern-Simons theory</a> which is <a class="existingWikiWord" href="/nlab/show/holographic+principle">dual</a> to the <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> on the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>. In this context it was first pointed out in (<a href="#Witten">Witten 1996</a>) and later elaborated on in (<a href="#HopkinsSinger">Hopkins-Singer</a>).</p> <p>Specifically, in this context <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is the 4-class of the <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle</a> underlying the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>, subject to the quantization condition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>4</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>a</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> G_4 = \frac{1}{2}(\frac{1}{2}p_1) + a \,, </annotation></semantics></math></div> <p>for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">a \in H^4(X, \mathbb{Z})</annotation></semantics></math>, which makes direct sense as an equation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4(X, \mathbb{Z})</annotation></semantics></math> if the <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> 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> happens to be such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}p_1</annotation></semantics></math> is further divisible by 2, and can be made sense of more generally in terms of <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> (which was suggested in (<a href="#Witten">Witten 1996</a>) and made precise sense of in (<a href="#HopkinsSinger">Hopkins-Singer</a>) ).</p> <p>For simplicity, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}p_1</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is further divisible by 2 in the following. We then may consider direct refinements of the above ingredients to <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> and so we consider <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">differential cocycles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>a</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><msup><mover><mi>H</mi><mo stretchy="false">^</mo></mover> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hat a, \hat G \in \hat H^4(X)</annotation></semantics></math> with</p> <div class="maruku-equation" id="eq:DifferentialQuantizationCondition"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mover><mi>a</mi><mo stretchy="false">^</mo></mover><mo>∈</mo><msup><mover><mi>H</mi><mo stretchy="false">^</mo></mover> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \hat G = \frac{1}{2}(\frac{1}{2}\hat \mathbf{p}_1) + \hat a \in \hat H^4(X) \,, </annotation></semantics></math></div> <p>where the differential refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{1}{2}\hat \mathbf{p}_1</annotation></semantics></math> is discussed in detail at <em><a class="existingWikiWord" href="/nlab/show/differential+string+structure">differential string structure</a></em>.</p> <p>Now, after <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">dimensional reduction</a> on a 4-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> on the remaining 7-dimensional <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> contains a <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher Chern-Simons term</a> which up to prefactors is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>↦</mo><mi>exp</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>∪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>−</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \hat G \mapsto \exp i \int_X ( \hat G \cup \hat G - (\frac{1}{4}\hat \mathbf{p}_1)^2 ) \,, </annotation></semantics></math></div> <p>where</p> <ul> <li> <p>the cup product now is the differential <a class="existingWikiWord" href="/nlab/show/Beilinson-Deligne+cup+product">Beilinson-Deligne cup product</a> refinement of the integral cup product;</p> </li> <li> <p>the symbol <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(i \int_X (-))</annotation></semantics></math> denotes <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a>.</p> </li> </ul> <p>Using <a class="maruku-eqref" href="#eq:DifferentialQuantizationCondition">(1)</a> this is</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><mi>exp</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><mrow><mo>(</mo><mover><mi>a</mi><mo stretchy="false">^</mo></mover><mo>∪</mo><mover><mi>a</mi><mo stretchy="false">^</mo></mover><mo>+</mo><mover><mi>a</mi><mo stretchy="false">^</mo></mover><mo>∪</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots = \exp i \int_X \left( \hat a \cup \hat a + \hat a \cup \frac{1}{2}\hat \mathbf{p}_1 \right) \,. </annotation></semantics></math></div> <p>But by corollary <a class="maruku-ref" href="#DivisibilityOfCupSquare"></a> this is further divisible by 2. Hence the generator of the group of higher Chern-Simons action functionals is one half of this</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>↦</mo><mi>exp</mi><mi>i</mi><msub><mo>∫</mo> <mi>X</mi></msub><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>∪</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>−</mo><mo stretchy="false">(</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \hat G \mapsto \exp i \int_X \frac{1}{2} ( \hat G \cup \hat G - (\frac{1}{4}\hat \mathbf{p}_1)^2 ) \,. </annotation></semantics></math></div> <p>In (<a href="#Witten">Witten 1996</a>) it is discussed that the space of <a class="existingWikiWord" href="/nlab/show/states">states</a> of this “fractional” functional over a 6-dimensional <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 the space of <a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a> of the <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a> on the <a class="existingWikiWord" href="/nlab/show/M5-brane">M5-brane</a>.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+Wu+structure">integral Wu structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Wu+structure">twisted Wu structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shifted+C-field+flux+quantization">shifted C-field flux quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontryagin+class">Pontryagin class</a>, <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+class">Stiefel-Whitney class</a>, <a class="existingWikiWord" href="/nlab/show/one-loop+anomaly+polynomial+I8">one-loop anomaly polynomial I8</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler+class">Euler class</a></p> </li> </ul> <h2 id="references">References</h2> <p>The original reference is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Wen-Tsun+Wu">Wen-Tsun Wu</a>, <em>On Pontrjagin classes: II</em> Sientia Sinica 4 (1955) 455-490</li> </ul> <p>See also around p. 228 of</p> <ul> <li id="MilnorStasheff74"><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Characteristic classes</em>, Princeton University Press (1974)</li> </ul> <p>and section 2 of</p> <ul> <li>Yanghyun Byun, <em>On vanishing of characteristic numbers in Poincaré complexes</em>, Transactions of the AMS, vol 348, number 8 (1996) (<a href="http://www.ams.org/journals/tran/1996-348-08/S0002-9947-96-01495-X/S0002-9947-96-01495-X.pdf">pdf</a>)</li> </ul> <p>and</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, Toshio Yoshida, <em>Wu classes</em> Proceedings of the American Mathematical Society Vol. 100, No. 2, (1987) (<a href="http://www.jstor.org/pss/2045970">JSTOR</a>)</li> </ul> <p>Details are reviewed in appendix E of</p> <ul id="HopkinsSinger"> <li><a class="existingWikiWord" href="/nlab/show/Mike+Hopkins">Mike Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Isadore+Singer">Isadore Singer</a>, <em><a class="existingWikiWord" href="/nlab/show/Quadratic+Functions+in+Geometry%2C+Topology%2C+and+M-Theory">Quadratic Functions in Geometry, Topology, and M-Theory</a></em></li> </ul> <p>This is based on or motivated from observations in</p> <ul id="Witten"> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Five-Brane Effective Action In M-Theory</em> (<a href="http://arxiv.org/abs/hep-th/9610234">arXiv:hep-th/9610234</a>)</li> </ul> <p>More discussion of Wu classes in this physical context is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Twisted topological structures related to M-branes II: Twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Wu</mi></mrow><annotation encoding="application/x-tex">Wu</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Wu</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Wu^c</annotation></semantics></math> structures</em> (<a href="http://arxiv.org/abs/1109.4461">arXiv:1109.4461</a>)</li> </ul> <p>which also summarizes many standard properties of Wu classes.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on November 11, 2019 at 07:51:25. See the <a href="/nlab/history/Wu+class" 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/Wu+class" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/3475/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/Wu+class/19" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Wu+class" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Wu+class" accesskey="S" class="navlink" id="history" rel="nofollow">History (19 revisions)</a> <a href="/nlab/show/Wu+class/cite" style="color: black">Cite</a> <a href="/nlab/print/Wu+class" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Wu+class" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>