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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="cobordism_theory">Cobordism theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a></strong> = <a class="existingWikiWord" href="/nlab/show/manifolds+and+cobordisms+-+contents">manifolds and cobordisms</a> + <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/equivariant+cobordism+theory">equivariant cobordism theory</a></li> </ul> <p><strong>Concepts of cobordism theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangential+structure">tangential structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>, <a class="existingWikiWord" href="/nlab/show/cobordism+class">cobordism class</a></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin%27s+theorem">Pontrjagin's theorem</a> (<a class="existingWikiWord" href="/nlab/show/equivariant+Pontrjagin+theorem">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Pontrjagin+theorem">twisted</a>):</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo>↔</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{\leftrightarrow}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+framed+submanifolds">normally framed submanifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+theorem">Thom's theorem</a>:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mo>↔</mo></mphantom></mrow><annotation encoding="application/x-tex">\phantom{\leftrightarrow}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of maps to <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> <a class="existingWikiWord" href="/nlab/show/MO">MO</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↔</mo></mrow><annotation encoding="application/x-tex">\leftrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cobordism+classes">cobordism classes</a> of <a class="existingWikiWord" href="/nlab/show/normally+oriented+submanifolds">normally oriented submanifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a></p> <p><a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a></p> <p><a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <p><a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+collapse+construction">Pontryagin-Thom collapse construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology+theory">complex cobordism cohomology theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> </li> </ul> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/bordism+homology+theories">bordism homology theories</a>/<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a>, their <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+rings">cobordism rings</a></strong>:</p> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/M%28B%2Cf%29">M(B,f)</a> (<a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MO">MO</a>, <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>, <a class="existingWikiWord" href="/nlab/show/MSpin">MSpin</a>, <a class="existingWikiWord" href="/nlab/show/MString">MString</a>, …</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MU">MU</a>, <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, …</p> <p><a class="existingWikiWord" href="/nlab/show/Ravenel%27s+spectrum">MΩΩSU(n)</a></p> <p><a class="existingWikiWord" href="/nlab/show/MP-theory">MP</a>, <a class="existingWikiWord" href="/nlab/show/MR-theory">MR</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSpin%5Ec">MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSp">MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/MOFr">MOFr</a>, <a class="existingWikiWord" href="/nlab/show/MUFr">MUFr</a>, <a class="existingWikiWord" href="/nlab/show/MSUFr">MSUFr</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bordism+homology+theory">equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MFr">equivariant MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MO">equivariant MO</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MU">equivariant MU</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+bordism+homology+theory">global equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mO">global equivariant mO</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mU">global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a></li> </ul> </div></div></div> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#the__spectrum'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">M U</annotation></semantics></math> spectrum</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#HomotopyGroups'>Homotopy groups: Cobordism and Lazard ring</a></li> <li><a href='#UniversalComplexOrientation'>Universal complex orientation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">M U</annotation></semantics></math></a></li> <li><a href='#homology_of_a_manifold_bordisms_in_'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-homology of a manifold: bordisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></a></li> <li><a href='#cohomology_of_a_manifold_cobordisms_in_'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-cohomology of a manifold: cobordisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></a></li> <li><a href='#homology_of__hopf_algebroid_structure_on_dual_steenrod_algebra'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-homology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>: Hopf algebroid structure on dual Steenrod algebra</a></li> <li><a href='#nilpotence_theorem'>Nilpotence theorem</a></li> <li><a href='#snaiths_theorem'>Snaith’s theorem</a></li> <li><a href='#localization_and_brownpeterson_spectrum'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-Localization and Brown-Peterson spectrum</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#differential_and_hodgefiltered_cobordism_cohomology'>Differential and Hodge-filtered cobordism cohomology</a></li> <li><a href='#relation_to_cft'>Relation to CFT</a></li> <li><a href='#RelationToDivisors'>Relation to divisors</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math> is the universal <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> for <a class="existingWikiWord" href="/nlab/show/complex+vector+bundles">complex vector bundles</a>. It is the spectrum <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <em>complex <a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></em>. It is the complex analog of <a class="existingWikiWord" href="/nlab/show/MO">MO</a>.</p> <p><a class="existingWikiWord" href="/nlab/show/MR+cohomology+theory">MR cohomology theory</a>, or <em>real cobordism</em>, (<a href="#Landweber68">Landweber 68</a>, <a href="#Landweber69">Landweber 69</a>) is the <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/equivariant+cohomology+theory">equivariant cohomology theory</a> version of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology+theory">complex cobordism cohomology theory</a>.</p> <h2 id="the__spectrum">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">M U</annotation></semantics></math> spectrum</h2> <p>The spectrum denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">M U</annotation></semantics></math> is, as a <a class="existingWikiWord" href="/nlab/show/sequential+spectrum">sequential spectrum</a>, in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2 n</annotation></semantics></math> given by the <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> of the underlying real vector bundle of the complex <a class="existingWikiWord" href="/nlab/show/universal+vector+bundle">universal vector bundle</a>: the <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> that is <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> by the defining <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^n</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Thom</mi><mrow><mo>(</mo><mi>standard</mi><mspace width="thickmathspace"></mspace><mi>associated</mi><mspace width="thickmathspace"></mspace><mi>bundle</mi><mspace width="thickmathspace"></mspace><mi>to</mi><mspace width="thickmathspace"></mspace><mi>universal</mi><mspace width="thickmathspace"></mspace><mi>bundle</mi><mrow><mtable><mtr><mtd><mi>E</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> M U(2n) = Thom \left( standard\;associated\;bundle\;to\;universal\;bundle \array{ E U(n) \\ \downarrow \\ B U(n) } \right) </annotation></semantics></math></div> <p>A priori this yields a <a href="Introduction+to+Stable+homotopy+theory+--+1#SequentialTSpectra">sequential S2-spectrum</a>, which is then turned into a sequential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>-spectrum by taking the component spaces in odd degree to be the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> of the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> with those in even degree.</p> <p>This represents a <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a> and indeed the universal one among these, see at <em><a class="existingWikiWord" href="/nlab/show/universal+complex+orientation+on+MU">universal complex orientation on MU</a></em>.</p> <p>The <em>periodic</em> complex cobordism theory is given by adding up all the even degree powers of this theory:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>P</mi><mo>=</mo><msub><mo>∨</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow></msub><msup><mi>Σ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mi>M</mi><mi>U</mi></mrow><annotation encoding="application/x-tex"> M P = \vee_{n \in \mathbb{Z}} \Sigma^{2 n} M U </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/cohomology+ring">cohomology ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>P</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M P({*})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard ring</a> which is the universal coefficient ring for <a class="existingWikiWord" href="/nlab/show/formal+group+laws">formal group laws</a>, see at <em><a class="existingWikiWord" href="/nlab/show/Milnor-Quillen+theorem+on+MU">Milnor-Quillen theorem on MU</a></em> .</p> <p>The <a class="existingWikiWord" href="/nlab/show/periodic+cohomology+theory">periodic</a> version is sometimes written <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PMU</mi></mrow><annotation encoding="application/x-tex">PMU</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="HomotopyGroups">Homotopy groups: Cobordism and Lazard ring</h3> <p>The <a class="existingWikiWord" href="/nlab/show/graded+ring">graded ring</a> given by evaluating complex cobordism theory on the point is both the complex <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a> as well as the <a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard ring</a> classifying <a class="existingWikiWord" href="/nlab/show/formal+group+laws">formal group laws</a>.</p> <div class="num_theorem" id="RelationToCobordismRing"> <h6 id="theorem">Theorem</h6> <p>Evaluation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math> on the point yields the complex <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a>, whose underlying group is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>*</mo></msub><mi>MU</mi><mo>≃</mo><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>pt</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \pi_\ast MU \simeq MU_\ast(pt) \simeq \mathbb{Z}[x_1, x_2, \cdots] \,, </annotation></semantics></math></div> <p>where the generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math> is in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>i</mi></mrow><annotation encoding="application/x-tex">2 i</annotation></semantics></math>.</p> </div> <p>This is due to (<a href="#Milnor60">Milnor 60</a>, <a href="#Novikov60">Novikov 60</a>, <a href="#Novikov62">Novikov 62</a>). A review is in (<a href="#Ravenel">Ravenel theorem 1.2.18</a>, <a href="#Ravenel">Ravenel, ch. 3, theorem 3.1.5</a>).</p> <p>The <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a> associated with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math> as with any <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a> is classified by a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>⟶</mo><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>MU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L \longrightarrow \pi_\bullet(MU)</annotation></semantics></math> out of the <a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard ring</a>.</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>This canonical <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> 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><mi>L</mi><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>MU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L \stackrel{\simeq}{\longrightarrow} \pi_\bullet(MU) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard ring</a> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology+ring">cohomology ring</a>, hence by theorem <a class="maruku-ref" href="#RelationToCobordismRing"></a> with the complex <a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a>.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Quillen%27s+theorem+on+MU">Quillen's theorem on MU</a>. (e.g <a href="#LurieLect7">Lurie 10, lect. 7, theorem 1</a>)</p> <h3 id="UniversalComplexOrientation">Universal complex orientation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">M U</annotation></semantics></math></h3> <p>There is a canonical <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex orientation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math> obtained from the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>:</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mover><mo>→</mo><mo>≃</mo></mover><mi>M</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>M</mi><mi>U</mi><mo stretchy="false">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega : \mathbb{C}P^\infty \stackrel{\simeq}{\to} M U(1) \;\;\;\; M U(\mathbb{C}P^\infty) </annotation></semantics></math></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> a <a class="existingWikiWord" href="/nlab/show/homotopy-commutative+ring+spectrum">homotopy-commutative ring spectrum</a> there is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> between <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex orientation</a> of <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 <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a> maps 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>MU</mi><mo>⟶</mo><mi>E</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> MU \longrightarrow E \,. </annotation></semantics></math></div> <p>(e.g <a href="#LurieLect6">Lurie 10, lect. 6, theorem 8</a>, <a href="#Ravenel">Ravenel, chapter 4, lemma 4.1.13</a>)</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/complex+orientation+and+MU">complex orientation and MU</a></em>.</p> <h3 id="homology_of_a_manifold_bordisms_in_"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-homology of a manifold: bordisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> or a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-homology <a class="existingWikiWord" href="/nlab/show/group">group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MU</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MU_\ast(X)</annotation></semantics></math> of its underlying <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> is the group of equivalence classes of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> from manifolds <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> with <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a> on the stable <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a>, modulo suitable complex <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a>.</p> <p>See <a href="#Ravenel">Ravenel chapter 1, section 2</a>.</p> <p>For more information, see the article <a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism homology theory</a>, which treats the oriented case; the case of (stable almost) complex structure is similar.</p> <h3 id="cohomology_of_a_manifold_cobordisms_in_"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-cohomology of a manifold: cobordisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></h3> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-cohomology groups of a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> can be expressed in terms of bordisms given by proper <a href="cobordism+cohomology+theory#ComplexOrientedMaps">complex-oriented maps</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>.</p> <p>For more information, see the article <em><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></em>.</p> <h3 id="homology_of__hopf_algebroid_structure_on_dual_steenrod_algebra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-homology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>: Hopf algebroid structure on dual Steenrod algebra</h3> <p>Moreover, the dual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MU</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>MU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MU_\bullet(MU)</annotation></semantics></math> forms a <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebroid">commutative Hopf algebroid</a> over the <a class="existingWikiWord" href="/nlab/show/Lazard+ring">Lazard ring</a>. This is the content of the <em><a class="existingWikiWord" href="/nlab/show/Landweber-Novikov+theorem">Landweber-Novikov theorem</a></em>.</p> <h3 id="nilpotence_theorem">Nilpotence theorem</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/nilpotence+theorem">nilpotence theorem</a></li> </ul> <h3 id="snaiths_theorem">Snaith’s theorem</h3> <p><a class="existingWikiWord" href="/nlab/show/Snaith%27s+theorem">Snaith's theorem</a> asserts that the <a class="existingWikiWord" href="/nlab/show/periodic+complex+cobordism+spectrum">periodic complex cobordism spectrum</a> is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+%E2%88%9E-ring">∞-group ∞-ring</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> for stable <a class="existingWikiWord" href="/nlab/show/complex+vector+bundles">complex vector bundles</a> (the classifying space for <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>) localized at the <a class="existingWikiWord" href="/nlab/show/Bott+element">Bott element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</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>PMU</mi><mo>≃</mo><mi>𝕊</mi><mo stretchy="false">[</mo><mi>B</mi><mi>U</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><msup><mi>β</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PMU \simeq \mathbb{S}[B U][\beta^{-1}] \,. </annotation></semantics></math></div> <h3 id="localization_and_brownpeterson_spectrum"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-Localization and Brown-Peterson spectrum</h3> <p>The <a class="existingWikiWord" href="/nlab/show/p-localization">p-localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math> decomposes into the <a class="existingWikiWord" href="/nlab/show/Brown-Peterson+spectra">Brown-Peterson spectra</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></li> </ul> <div> <p><strong>flavors of <a class="existingWikiWord" href="/nlab/show/bordism+homology+theories">bordism homology theories</a>/<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theories">cobordism cohomology theories</a>, their <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <a class="existingWikiWord" href="/nlab/show/Thom+spectra">Thom spectra</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+rings">cobordism rings</a></strong>:</p> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/M%28B%2Cf%29">M(B,f)</a> (<a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/MFr">MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MO">MO</a>, <a class="existingWikiWord" href="/nlab/show/MSO">MSO</a>, <a class="existingWikiWord" href="/nlab/show/MSpin">MSpin</a>, <a class="existingWikiWord" href="/nlab/show/MString">MString</a>, …</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MU">MU</a>, <a class="existingWikiWord" href="/nlab/show/MSU">MSU</a>, …</p> <p><a class="existingWikiWord" href="/nlab/show/Ravenel%27s+spectrum">MΩΩSU(n)</a></p> <p><a class="existingWikiWord" href="/nlab/show/MP-theory">MP</a>, <a class="existingWikiWord" href="/nlab/show/MR-theory">MR</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSpin%5Ec">MSpin<sup><i>c</i></sup></a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MSp">MSp</a></p> </li> </ul> <p>relative bordism theories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/MOFr">MOFr</a>, <a class="existingWikiWord" href="/nlab/show/MUFr">MUFr</a>, <a class="existingWikiWord" href="/nlab/show/MSUFr">MSUFr</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bordism+homology+theory">equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MFr">equivariant MFr</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MO">equivariant MO</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+MU">equivariant MU</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+bordism+homology+theory">global equivariant bordism theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mO">global equivariant mO</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+mU">global equivariant mU</a></p> </li> </ul> <p>algebraic:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+cobordism">algebraic cobordism</a></li> </ul> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li id="Novikov62"> <p><a class="existingWikiWord" href="/nlab/show/Sergei+Novikov">Sergei Novikov</a>, <em>Homotopy properties of Thom complexes</em>, Mat. Sbornik 57 (1962), no. 4, 407–442, 407–442 (<a href="http://www.mi-ras.ru/~snovikov/6.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/NovikovThomComplexes.pdf" title="pdf">pdf</a>)</p> </li> <li id="ConnerFloyd66"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Edwin+Floyd">Edwin Floyd</a>, Section 12 of: <em><a class="existingWikiWord" href="/nlab/show/The+Relation+of+Cobordism+to+K-Theories">The Relation of Cobordism to K-Theories</a></em>, Lecture Notes in Mathematics <strong>28</strong> Springer 1966 (<a href="https://link.springer.com/book/10.1007/BFb0071091">doi:10.1007/BFb0071091</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=216511">MR216511</a>)</p> </li> <li id="Stong68"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Stong">Robert Stong</a>, Chapter VII of: <em>Notes on Cobordism theory</em>, Princeton University Press, 1968 (<a href="http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf">toc pdf</a>, <a href="http://press.princeton.edu/titles/6465.html">ISBN:9780691649016</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf">pdf</a>)</p> </li> <li id="ConnerSmith69"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Larry+Smith">Larry Smith</a>, <em>On the complex bordism of finite complexes</em>, Publications Mathématiques de l’IHÉS, Tome 37 (1969) , pp. 117-221 (<a href="http://www.numdam.org/item/?id=PMIHES_1969__37__117_0">numdam:PMIHES_1969__37__117_0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Larry+Smith">Larry Smith</a>, <em>On Realizing Complex Bordism Modules: Applications to the Stable Homotopy of Spheres</em>, American Journal of Mathematics Vol. 92, No. 4 (Oct., 1970), pp. 793-856 (<a href="https://doi.org/10.2307/2373397">doi:10.2307/2373397</a>)</p> </li> <li id="Landweber70"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Landweber">Peter Landweber</a>, <em>On the complex bordism and cobordism of infinite complexes</em>, Bull. Amer. Math. Soc. Volume 76, Number 3 (1970) (<a href="http://projecteuclid.org/euclid.bams/1183531832">Euclid</a>)</p> </li> <li id="Quillen71"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, <em>Elementary Proofs of Some Results of Cobordism Theory Using Steenrod Operations</em>, Advances in Mathematics <strong>7</strong> (1971) 29–56 [<a href="http://dx.doi.org/10.1016/0001-8708(71)90041-7">doi:10.1016/0001-8708(71)90041-7</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Frank+Adams">John Frank Adams</a>, <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalized+homology">Stable homotopy and generalized homology</a></em>, Chicago Lectures in Mathematics, The University of Chicago Press (1974) [<a href="https://www.press.uchicago.edu/ucp/books/book/chicago/S/bo21302708.html">ucp:bo21302708</a>, <a href="https://www.uio.no/studier/emner/matnat/math/MAT9580/v17/documents/adams-shgh.pdf">pdf</a>]</p> </li> <li id="Ravenel"> <p><a class="existingWikiWord" href="/nlab/show/Doug+Ravenel">Doug Ravenel</a>, <em><a class="existingWikiWord" href="/nlab/show/Complex+cobordism+and+stable+homotopy+groups+of+spheres">Complex cobordism and stable homotopy groups of spheres</a></em> (1986)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</a>, Section 4.4 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>, Fields Institute Monographs, American Mathematical Society, 1996 (<a href="https://cdsweb.cern.ch/record/2264210">cds:2264210</a>)</p> </li> <li id="EKMM97"> <p><a class="existingWikiWord" href="/nlab/show/Anthony+Elmendorf">Anthony Elmendorf</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Mandell">Michael Mandell</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, section VIII of <em><a class="existingWikiWord" href="/nlab/show/Rings%2C+modules+and+algebras+in+stable+homotopy+theory">Rings, modules and algebras in stable homotopy theory</a></em> 1997 (<a href="http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf">pdf</a>)</p> </li> <li id="Hopkins99"> <p><a class="existingWikiWord" href="/nlab/show/Mike+Hopkins">Mike Hopkins</a>, section 4 of <em>Complex oriented cohomology theories and the language of stacks</em>, course notes 1999 (<a href="http://www.math.rochester.edu/u/faculty/doug/otherpapers/coctalos.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dai+Tamaki">Dai Tamaki</a>, <a class="existingWikiWord" href="/nlab/show/Akira+Kono">Akira Kono</a>, Section 3.7 and Chapter 6 in: <em>Generalized Cohomology</em>, Translations of Mathematical Monographs, American Mathematical Society, 2006 (<a href="https://bookstore.ams.org/mmono-230">ISBN: 978-0-8218-3514-2</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Chromatic+Homotopy+Theory">Chromatic Homotopy Theory</a></em> Lecture series (<a href="http://www.math.harvard.edu/~lurie/252x.html">lecture notes</a>), Lecture 5 <em>Complex bordism</em></p> <p>(<a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture5.pdf">pdf</a>)</p> </li> <li id="LurieLect6"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Chromatic+Homotopy+Theory">Chromatic Homotopy Theory</a></em> Lecture series (<a href="http://www.math.harvard.edu/~lurie/252x.html">lecture notes</a>) Lecture 6 <em>MU and complex orientations</em> (<a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture6.pdf">pdf</a>)</p> </li> <li id="LurieLect7"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Chromatic+Homotopy+Theory">Chromatic Homotopy Theory</a></em> Lecture series (<a href="http://www.math.harvard.edu/~lurie/252x.html">lecture notes</a>), Lecture 7 <em>The homology of MU</em> (<a href="http://www.math.harvard.edu/~lurie/252xnotes/Lecture7.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Manifold+Atlas">Manifold Atlas</a>, <em><a href="http://www.map.mpim-bonn.mpg.de/Complex_bordism">Complex bordism</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>Products on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MU</mi></mrow><annotation encoding="application/x-tex">MU</annotation></semantics></math>-modules</em> (<a href="http://hopf.math.purdue.edu/Strickland/mult.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jesse+McKeown">Jesse McKeown</a>, <em>Complex Cobordism vs. Representing Formal Group Laws</em> (<a href="http://arxiv.org/abs/1605.09252">arXiv:1605.09252</a>)</p> </li> </ul> <p>For general discussion of equivariant complex oriented cohomology see at <em><a href="equivariant+cohomology#InComplexOrientedGeneralizedCohomologyTheory">equivariant cohomology – References – Complex oriented cohomology</a></em></p> <p>On the <a class="existingWikiWord" href="/nlab/show/Chern-Dold+character">Chern-Dold character</a> on complex cobordism:</p> <ul> <li id="Buchstaber70"> <p><a class="existingWikiWord" href="/nlab/show/Victor+Buchstaber">Victor Buchstaber</a>, <em>The Chern–Dold character in cobordisms. I</em>,</p> <p>Russian original: Mat. Sb. (N.S.), 1970 Volume 83(125), Number 4(12), Pages 575–595 (<a href="http://m.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=3530&option_lang=eng">mathnet:3530</a>)</p> <p>English translation: Mathematics of the USSR-Sbornik, Volume 12, Number 4, AMS 1970 (<a href="https://iopscience.iop.org/article/10.1070/SM1970v012n04ABEH000939">doi:10.1070/SM1970v012n04ABEH000939</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Victor+Buchstaber">Victor Buchstaber</a>, A. P. Veselov, <em>Chern-Dold character in complex cobordisms and abelian varieties</em> (<a href="https://arxiv.org/abs/2007.05782">arXiv:2007.05782</a>)</p> </li> </ul> <h3 id="differential_and_hodgefiltered_cobordism_cohomology">Differential and Hodge-filtered cobordism cohomology</h3> <p>On <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a> (enhancement of cobordism cohomology to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Schick">Thomas Schick</a>, Ingo Schroeder, Moritz Wiethaup, <em>Landweber exact formal group laws and smooth cohomology theories</em>, Algebr. Geom. Topol. <strong>9</strong> (2009) 1751-1790 [<a href="https://arxiv.org/abs/0711.1134">arXiv:0711.1134</a>, <a href="https://doi.org/10.2140/agt.2009.9.1751">doi:10.2140/agt.2009.9.1751</a>]</li> </ul> <p>The notion of <a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge filtered differential</a> complex cobordism theory:</p> <ul> <li id="HopkinsQuick14"><a class="existingWikiWord" href="/nlab/show/Michael+J.+Hopkins">Michael J. Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, §5 in: <em>Hodge filtered complex bordism</em>, Journal of Topology <strong>8</strong> 1 (2014) 147-183 [<a href="https://arxiv.org/abs/1212.2173">arXiv:1212.2173</a>, <a href="https://doi.org/10.1112/jtopol/jtu021">doi:10.1112/jtopol/jtu021</a>]</li> </ul> <p>Introduction and survey:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, <em>Geometric Hodge filtered complex cobordism</em>, <a href="Center+for+Quantum+and+Topological+Systems#QuickMar2023">talk at</a> <em><a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a></em> (March 2023) [video:<a href="https://www.youtube.com/watch?v=pMu0gT5kIBo">YT</a>]</li> </ul> <p>A geometric cocycle model:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Knut+B.+Haus">Knut B. Haus</a>, <a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, <em>Geometric Hodge filtered complex cobordism</em> [<a href="https://arxiv.org/abs/2210.13259">arXiv:2210.13259</a>]</li> </ul> <p>Refinement of the <a class="existingWikiWord" href="/nlab/show/Abel-Jacobi+map">Abel-Jacobi map</a> to <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtered</a> <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology+theory">differential</a> <a class="existingWikiWord" href="/nlab/show/MU">MU</a>-<a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, <em>An Abel-Jacobi invariant for cobordant cycles</em>, Documenta Mathematica <strong>21</strong> (2016) 1645–1668 [<a href="https://arxiv.org/abs/1503.08449">arXiv:1503.08449</a>]</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/Umkehr+maps">Umkehr maps</a> in this context:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Knut+Bjarte+Haus">Knut Bjarte Haus</a>, <a class="existingWikiWord" href="/nlab/show/Gereon+Quick">Gereon Quick</a>, <em>Geometric pushforward in Hodge filtered complex cobordism and secondary invariants</em> [<a href="https://arxiv.org/abs/2303.15899">arXiv:2303.15899</a>]</li> </ul> <h3 id="relation_to_cft">Relation to CFT</h3> <p>A relation to <a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a> over <a class="existingWikiWord" href="/nlab/show/Spec%28Z%29">Spec(Z)</a> was suggested in</p> <ul> <li>Toshiyuki Katsura, Yuji Shimizu, <a class="existingWikiWord" href="/nlab/show/Kenji+Ueno">Kenji Ueno</a>, <em>Complex cobordism ring and conformal field theory over <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></em>, Mathematische Annalen March 1991, Volume 291, Issue 1, pp 551-571 (<a href="http://link.springer.com/article/10.1007%2FBF01445226">journal</a>)</li> </ul> <h3 id="RelationToDivisors">Relation to divisors</h3> <p>Relation of <a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a> with <a class="existingWikiWord" href="/nlab/show/divisors">divisors</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+cycles">algebraic cycles</a> and <a class="existingWikiWord" href="/nlab/show/Chow+groups">Chow groups</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Burt+Totaro">Burt Totaro</a>, <em>Torsion algebraic cycles and complex cobordism</em>, J. Amer. Math. Soc. 10 (1997), 467-493 (<a href="https://doi.org/10.1090/S0894-0347-97-00232-4">doi:10.1090/S0894-0347-97-00232-4</a>)</p> </li> <li> <p><a href="https://mathoverflow.net/a/272131/381">MO discussion</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 9, 2023 at 17:45:22. See the <a href="/nlab/history/MU" 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/MU" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/11727/#Item_7">Discuss</a><span class="backintime"><a href="/nlab/revision/MU/37" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/MU" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/MU" accesskey="S" class="navlink" id="history" rel="nofollow">History (37 revisions)</a> <a href="/nlab/show/MU/cite" style="color: black">Cite</a> <a href="/nlab/print/MU" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/MU" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>