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href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="representation_theory">Representation theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+object">equivariant object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/socle">socle</a>, <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+algebra">module algebra</a>, <a class="existingWikiWord" href="/nlab/show/comodule+algebra">comodule algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+action">Hopf action</a>, <a class="existingWikiWord" href="/nlab/show/measuring">measuring</a></p> </li> </ul> <h2 id="geometric_representation_theory">Geometric representation theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/perverse+sheaf">perverse sheaf</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, <a class="existingWikiWord" href="/nlab/show/lambda-ring">lambda-ring</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+function">symmetric function</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>, <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>, <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstruction+theorems">reconstruction theorems</a></p> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> </li> <li> <p><span class="newWikiWord">Be?linson-Bernstein localization<a href="/nlab/new/Be%3Flinson-Bernstein+localization">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/representation+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#presentations'>Presentations</a></li> <li><a href='#Borel'>Borel equivariant cohomology</a></li> <ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#group_cohomology'>Group cohomology</a></li> <li><a href='#equivariant_cohomotopy'>Equivariant cohomotopy</a></li> <li><a href='#equivariant_bundles'>Equivariant bundles</a></li> <li><a href='#local_systems__flat_connections'>Local systems – flat connections</a></li> <li><a href='#equivariant_de_rham_cohomology'>Equivariant de Rham cohomology</a></li> </ul> <li><a href='#remarks'>Remarks</a></li> </ul> <li><a href='#Bredon'>Bredon equivariant cohomology</a></li> <ul> <li><a href='#preliminary_remarks'>Preliminary remarks</a></li> <li><a href='#equivariant_spectra'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant spectra</a></li> <li><a href='#examples_2'>Examples</a></li> </ul> <li><a href='#multiplicative_equivariant_cohomology'>Multiplicative equivariant cohomology</a></li> <li><a href='#examples_3'>Examples</a></li> <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='#InComplexOrientedGeneralizedCohomologyTheory'>In complex oriented generalized cohomology theory</a></li> <li><a href='#traditional_orbifold_cohomology'>Traditional orbifold cohomology</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p><em>Equivariant cohomology</em> is <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in the presence of and taking into account <a class="existingWikiWord" href="/nlab/show/group">group</a>-<a class="existingWikiWord" href="/nlab/show/actions">actions</a> (and generally <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-actions">∞-actions</a>) both on the domain space and on the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>.</p> <p>Exactly what this comes down to depends on the choice of ambient <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> and of the way that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is regarded as an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. Some important choices are the following:</p> <ul> <li> <p><strong>“coarse” equivariance.</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msub><mi>L</mi> <mi>whe</mi></msub></mrow><annotation encoding="application/x-tex">\simeq L_{whe}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, regarded via its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid/<a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}G \in \mathbf{H}</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by the <a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a> on the category of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> equipped with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action. (This is also called the <em>coarse</em> <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a>, in view of the next examples). This theory only knows <a class="existingWikiWord" href="/nlab/show/homotopy+quotients">homotopy quotients</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+fixed+points">homotopy fixed points</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> (in particular cofibrant replacement in the <a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a> is indeed given by the <a class="existingWikiWord" href="/nlab/show/Borel+construction">Borel construction</a> and so Borel equivariant cohomology theory appears here whenever the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> have trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action). In the case that the domain itself is the points with trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-action then the equivariant cohomology here is precisely the <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </li> <li> <p><strong>“fine” Bredon equivariance.</strong> In order to bring in more geometric information one may equip <a class="existingWikiWord" href="/nlab/show/G-spaces">G-spaces</a> with information about the actual <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fixed+points">fixed points</a>, not just their <a class="existingWikiWord" href="/nlab/show/homotopy+fixed+points">homotopy fixed points</a>. By general lore of <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> this means to have all spaces be probe-able by fixed points, hence to have them be <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaves">(∞,1)-presheaves</a> on the <a class="existingWikiWord" href="/nlab/show/global+equivariant+indexing+category">global equivariant indexing category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Glob</mi></mrow><annotation encoding="application/x-tex">Glob</annotation></semantics></math>, or if desired just on the <a class="existingWikiWord" href="/nlab/show/global+orbit+category">global orbit category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Orb</mi></mrow><annotation encoding="application/x-tex">Orb</annotation></semantics></math>, hence to set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>Glob</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>Orb</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{H} \to \mathbf{B}) = (PSh_\infty(Glob) \to PSh_\infty(Orb))</annotation></semantics></math>, where the <a class="existingWikiWord" href="/nlab/show/base+%28%E2%88%9E%2C1%29-topos">base (∞,1)-topos</a> is that of <a class="existingWikiWord" href="/nlab/show/orbispaces">orbispaces</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> sitting <a class="existingWikiWord" href="/nlab/show/cohesion">cohesively</a> over it is the “<a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a>” proper (see there).</p> <p>Now we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}G \in \mathbf{H}</annotation></semantics></math> naturally via the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+embedding">(∞,1)-Yoneda embedding</a> and the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mo>≃</mo><msub><mi>L</mi> <mi>fpwe</mi></msub><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}_{/\mathbf{B}G} \simeq L_{fpwe} G Top </annotation></semantics></math> is the traditional <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a> presented by the “fine” model structure on <a class="existingWikiWord" href="/nlab/show/G-spaces">G-spaces</a> whose weak equivalences are the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-fixed point wise <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> for all suitable subgroups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> here are what are called <a class="existingWikiWord" href="/nlab/show/spectra+with+G-action">spectra with G-action</a> or “<a class="existingWikiWord" href="/nlab/show/naive+G-spectra">naive G-spectra</a>”. See at <em><a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a></em> for details. By the discussion there every object in the fine model structure if fibrant and cofibrant replacement here is given by passage to <a class="existingWikiWord" href="/nlab/show/G-CW+complexes">G-CW complexes</a>, so that the <a class="existingWikiWord" href="/nlab/show/derived+hom+spaces">derived hom spaces</a> computing cohomology are the ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fixed+points">fixed points</a> of the <a class="existingWikiWord" href="/nlab/show/mapping+spectra">mapping spectra</a> from such as <a class="existingWikiWord" href="/nlab/show/G-CW+complex">G-CW complex</a> into the coefficient spectrum (this is traditionally motivated via detour through <a class="existingWikiWord" href="/nlab/show/genuine+G-spectra">genuine G-spectra</a>, see e.f. <a href="#GreenleesMay">Greenlees-May, equation (3.7)</a>).</p> <p>Cohomology with <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+object">Eilenberg-MacLane object</a>-<a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>Orb</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">PSh_\infty(Orb)_{/\mathbf{B}G}</annotation></semantics></math> is what <a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a> originally considered as what is now called <em><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></em>.</p> </li> </ul> <h2 id="presentations">Presentations</h2> <blockquote> <p>under construction</p> </blockquote> <p>(…) <a class="existingWikiWord" href="/nlab/show/Elmendorf+theorem">Elmendorf theorem</a> (…) <a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a> (…)</p> <h2 id="Borel">Borel equivariant cohomology</h2> <p>We first state the <a class="existingWikiWord" href="/nlab/show/category+theory">general abstract</a> definition of <em><a class="existingWikiWord" href="/nlab/show/Borel+equivariant+cohomology">Borel equivariant cohomology</a></em> and then derive from it the more concrete formulations that are traditionally given in the literature.</p> <div class="standout"> <p><a class="existingWikiWord" href="/nlab/show/Borel+equivariant+cohomology">Borel equivariant cohomology</a> is the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <a class="existingWikiWord" href="/nlab/show/action+groupoids">action groupoids</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+quotients">homotopy quotients</a>/<a class="existingWikiWord" href="/nlab/show/Borel+constructions">Borel constructions</a>).</p> </div> <p>For standard <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> these <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a>s of a <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><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> acting on a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are traditionally known as the <a class="existingWikiWord" href="/nlab/show/Borel+construction">Borel construction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mi>G</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}G \times_G X</annotation></semantics></math>.</p> <p>Recall from the discussion at <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> that in full generality we have a notion of cohomology of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with coefficients in an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> are objects of some <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. The cohomology set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X,A)</annotation></semantics></math> is the set of connected components in the <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X,A) = \pi_0 \mathbf{H}(X,A)</annotation></semantics></math>.</p> <p>Recall moreover from the discussion at <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a> that objects of an <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves">(∞,1)-topos of (∞,1)-sheaves</a> have the interpretation of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s with extra structure. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves on a <a class="existingWikiWord" href="/nlab/show/site">site</a> of smooth test spaces such as <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> these objects have the interpretation of <a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-groupoid">Lie ∞-groupoid</a>s.</p> <p>In this case, 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> some such <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> with structure, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_0 \hookrightarrow X</annotation></semantics></math> be its 0-truncation, which is the <a class="existingWikiWord" href="/nlab/show/space">space</a> of <a class="existingWikiWord" href="/nlab/show/object">object</a>s 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>, the <a class="existingWikiWord" href="/nlab/show/discrete+groupoid">categorically discrete groupoid</a> underlying <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 think of the morphisms 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> as determining which points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> are related under some kind of action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math>, the 2-morphisms as relating these relations on some higher action, and so on. <strong>Equivariance</strong> means, roughly: functorial transformation behaviour of objects on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> with respect to this “action” encoded in the morphisms 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>. This is the intuition that is made precise in the following</p> <p>In the simple special case that one should keep in mind, <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 for instance the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X = X_0//G</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/action">action</a>, in the ordinary sense, of a <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><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math>: its morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>→</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \to g(x)</annotation></semantics></math> connect those objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> that are related by the action by some group element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math>.</p> <p>It is natural to consider the <a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a> of the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_0 \hookrightarrow X</annotation></semantics></math>. Equivariant cohomology is essentially just another term for relative cohomology with respect to an inclusion of a space into a (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-)groupoid.</p> <div class="un_defn"> <h6 id="definition_equivariant_cohomology">Definition (equivariant cohomology)</h6> <p>In some <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> the <strong>equivariant cohomology</strong> with coefficient in an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of a 0-truncated object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> with respect to an action encoded in an inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_0 \hookrightarrow X</annotation></semantics></math> is simply the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-valued cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X,A)</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>.</p> <p>More specifically, an <strong>equivariant structure</strong> on an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-cocycle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">c : X_0 \to A</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> is a choice of extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>c</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat c</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>0</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mover><mi>c</mi><mo stretchy="false">^</mo></mover></msub></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X_0 &\to& A \\ \downarrow & \nearrow_{\hat c} \\ X } \,. </annotation></semantics></math></div> <p>i.e. a lift of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> through the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X,A) \to \mathbf{H}(X_0,A)</annotation></semantics></math>.</p> </div> <h3 id="examples">Examples</h3> <h4 id="group_cohomology">Group cohomology</h4> <p>By comparing the definition of equivariant cohomology with that of <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> one sees that group cohomology can be equivalently thought of as being <strong>equivariant cohomology of the point</strong>.</p> <h4 id="equivariant_cohomotopy">Equivariant cohomotopy</h4> <div> <table><thead><tr><th>flavours of <br /><strong><a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></strong> <br /><a class="existingWikiWord" href="/nlab/show/generalized+cohomology">cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> <br /> (<a class="existingWikiWord" href="/nlab/show/homotopy+theory">full</a> or <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>)</th><th><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> <br /> (<a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">full</a> or <a class="existingWikiWord" href="/nlab/show/rational+equivariant+homotopy+theory">rational</a>)</th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/non-abelian+cohomology">non-abelian cohomology</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a> <br /> (full or <a class="existingWikiWord" href="/nlab/show/rational+Cohomotopy">rational</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivariant+Cohomotopy">equivariant Cohomotopy</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></strong> <br /> (<strong><a class="existingWikiWord" href="/nlab/show/parametrized+homotopy+theory">full</a></strong> or <strong><a class="existingWikiWord" href="/nlab/show/rational+parameterized+stable+homotopy+theory">rational</a></strong>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+Cohomotopy">twisted Cohomotopy</a></td><td style="text-align: left;">twisted equivariant Cohomotopy</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">stable cohomology</a></strong> <br /> (<strong><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">full</a></strong> or <strong><a class="existingWikiWord" href="/nlab/show/rational+stable+homotopy+theory">rational</a></strong>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/stable+Cohomotopy">stable Cohomotopy</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivariant+stable+Cohomotopy">equivariant stable Cohomotopy</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+Cohomotopy">differential Cohomotopy</a></td><td style="text-align: left;">equivariant differential cohomotopy</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/persistent+homotopy">persistent cohomology</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/persistent+Cohomotopy">persistent Cohomotopy</a></td><td style="text-align: left;">persistent equivariant Cohomotopy</td></tr> </tbody></table> </div> <h4 id="equivariant_bundles">Equivariant bundles</h4> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> some <a class="existingWikiWord" href="/nlab/show/group">group</a> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi></mrow><annotation encoding="application/x-tex">G Bund</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/stack">stack</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> be some finite group (just for the sake of simplicity of the example) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K \to Aut(X_0)</annotation></semantics></math> be an action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> on a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">X = X_0 // K</annotation></semantics></math> be the corresponding <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a>.</p> <p>Then a cocycle in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>-equivariant cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>K</mi><mo>,</mo><mi>G</mi><mi>Bund</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H(X_0//K, G Bund)</annotation></semantics></math> is</p> <ul> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">k \in K</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mi>k</mi></msub><mo>:</mo><mi>P</mi><mo>→</mo><msup><mi>k</mi> <mo>*</mo></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">\lambda_k : P \to k^* P</annotation></semantics></math></p> </li> <li> <p>such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>2</mn></msub><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">k_1, k_2 \in K</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></msub><mo>∘</mo><msub><mi>λ</mi> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msub><mo>=</mo><msub><mi>λ</mi> <mrow><msub><mi>k</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mi>k</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\lambda_{k_2}\circ \lambda_{k_1} = \lambda_{k_2\cdot k_1}</annotation></semantics></math>.</p> </li> </ul> <h4 id="local_systems__flat_connections">Local systems – flat connections</h4> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/space">space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><mo>=</mo><msub><mi>P</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X := P_n(X_0)</annotation></semantics></math> a version of its <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a> we have a canonical inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><msub><mi>P</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X_0 \hookrightarrow P_n(X_0)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> as the collection of constant paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math>.</p> <p>Consider for definiteness <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><msub><mi>Π</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X_0) := \Pi_\infty(X_0)</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/path+%E2%88%9E-groupoid">path ∞-groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math>. (All other cases are in principle obtaind from this by truncation and/or strictification).</p> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> some coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">g : X_0 \to A</annotation></semantics></math> can be thought of as classifying a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> on the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math>.</p> <p>On the other hand, a morphism out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_n(X_0)</annotation></semantics></math> is something like a flat <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> (see there for more details) on this principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle, also called an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+system">local system</a>. (More details on this are at <span class="newWikiWord">differential cohomology<a href="/schreiber/new/differential+cohomology">?</a></span>).</p> <p>Accordingly, an extension of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">g : X_0 \to A</annotation></semantics></math> through the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X_0 \hookrightarrow \Pi(X)</annotation></semantics></math> is the process of equipping a principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle with a flat connection.</p> <p>Comparing with the above definition of eqivariant cohomology, we see that flat connections on bundles may be regarded as <strong>path-equivariant structures</strong> on these bundles.</p> <p>This is therefore an example of equivariance which is not with respect to a global <a class="existingWikiWord" href="/nlab/show/group">group</a> action, but genuinely a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>al one.</p> <h4 id="equivariant_de_rham_cohomology">Equivariant de Rham cohomology</h4> <ul> <li><a class="existingWikiWord" href="/nlab/show/equivariant+de+Rham+cohomology">equivariant de Rham cohomology</a></li> </ul> <h3 id="remarks">Remarks</h3> <p>When pairing equivariant cohomology with other variants of cohomology such as <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> or <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> one has to exercise a bit of care as to what it really is that one wants to consider. A discussion of this is (beginning to appear) at <a class="existingWikiWord" href="/schreiber/show/differential+equivariant+cohomology">differential equivariant cohomology</a>.</p> <h2 id="Bredon">Bredon equivariant cohomology</h2> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></li> </ul> <h3 id="preliminary_remarks">Preliminary remarks</h3> <p>According to the <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a> on <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> are objects in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, the 0th cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^0(X;A)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(Map(X,A))</annotation></semantics></math>, while if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">group object</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1(X;A)= \pi_0(Map(X,B A))</annotation></semantics></math>. More generally, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is <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> times <a class="existingWikiWord" href="/nlab/show/delooping">deloopable</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>B</mi> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^n(X;A) = \pi_0(Map(X, B^n A)</annotation></semantics></math>. In <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, this gives you the usual notions if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a (discrete) group, and in general, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1(X;A)</annotation></semantics></math> classifies <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s in whatever <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>.</p> <p>Now consider the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">G Top</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant spaces, which can also be described as the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaves</a> on the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. For any other group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> there is a notion of a principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>Π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G,\Pi)</annotation></semantics></math>-bundle (where <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 group of equivariance, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> is the structure group of the bundle), and these are classified by maps into a classifying <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>-space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>G</mi></msub><mi>Π</mi></mrow><annotation encoding="application/x-tex">B_G \Pi</annotation></semantics></math>. So the principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>Π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G,\Pi)</annotation></semantics></math>-bundles over <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> can be called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><msub><mi>B</mi> <mi>G</mi></msub><mi>Π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^0(X;B_G \Pi)</annotation></semantics></math>. If we had something of which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>G</mi></msub><mi>Π</mi></mrow><annotation encoding="application/x-tex">B_G \Pi</annotation></semantics></math> was a <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>, we could call the principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>Π</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G,\Pi)</annotation></semantics></math>-bundles “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mo>?</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^1(X;?)</annotation></semantics></math>”, but there does not seem to be such a thing. It seems that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>G</mi></msub><mi>Π</mi></mrow><annotation encoding="application/x-tex">B_G \Pi</annotation></semantics></math> is not connected, in the sense that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><msub><mi>B</mi> <mi>G</mi></msub><mi>Π</mi></mrow><annotation encoding="application/x-tex">{*}\to B_G \Pi</annotation></semantics></math> is not an <a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a> and thus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>G</mi></msub><mi>Π</mi></mrow><annotation encoding="application/x-tex">B_G \Pi</annotation></semantics></math> is not the quotient of a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">group object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">G Top</annotation></semantics></math>.</p> <h3 id="equivariant_spectra"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant spectra</h3> <p>If we have an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of our <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos that can be delooped infinitely many times, then we can define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^n(X;A)</annotation></semantics></math> for any integer <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> by looking at all the spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup><mi>A</mi><mo>=</mo><msup><mi>B</mi> <mi>n</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\Omega^{-n} A = B^n A</annotation></semantics></math>. These integer-graded <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a>s are closely connected to each other, e.g. they often have <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a>s or <a class="existingWikiWord" href="/nlab/show/Steenrod+square">Steenrod square</a>s or <a class="existingWikiWord" href="/nlab/show/Poincare+duality">Poincare duality</a>, so it makes sense to consider them all together as a <em><a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a></em> . We then are motivated to put together all of the objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>B</mi> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{B^n A\}</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a>, a single object which encodes all of the cohomology groups of the theory. A general spectrum is a sequence of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E_n\}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub><mo>≃</mo><mi>Ω</mi><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">E_n \simeq \Omega E_{n+1}</annotation></semantics></math>; the stronger requirement that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≃</mo><mi>B</mi><msub><mi>E</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">E_{n+1} \simeq B E_n</annotation></semantics></math> restricts us to “connective” spectra, those that can be produced by successively delooping a single object of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos. In <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, the most “basic” spectra are the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectra</a> produced from the input of an ordinary abelian group.</p> <p>Now we can do all of this in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">G Top</annotation></semantics></math>, and the resulting notion of spectrum is called a <strong><a class="existingWikiWord" href="/nlab/show/naive+G-spectrum">naive G-spectrum</a></strong>: a sequence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{E_n\}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>n</mi></msub><mo>≃</mo><mi>Ω</mi><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">E_n \simeq \Omega E_{n+1}</annotation></semantics></math>. Any naive <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>-spectrum represents a cohomology theory on <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>-spaces. The most “basic” of these are “Eilenberg-Mac Lane <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>-spectra” produced from <strong><a class="existingWikiWord" href="/nlab/show/coefficient+system">coefficient systems</a></strong>, i.e. abelian-group-valued presheaves on the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a>. The cohomology theory represented by such an Eilenberg-Mac Lane <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>-spectrum is called an (integer-graded) <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a> theory.</p> <p>It turns out, though, that the cohomology theories arising in this way are kind of weird. For instance, when one calculates with them, one sees <a class="existingWikiWord" href="/nlab/show/torsion">torsion</a> popping up in odd places where one wouldn’t expect it. It would also be nice to have a <a class="existingWikiWord" href="/nlab/show/Poincare+duality">Poincare duality</a> theorem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-manifolds, but that fails with these theories. The solution people have come up with is to widen the notion of “<a class="existingWikiWord" href="/nlab/show/loop+space+object">looping</a>” and “<a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>” and thereby the grading:</p> <p>instead of just looking at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo>=</mo><mi>Map</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^n = Map(S^n, -)</annotation></semantics></math>, we look at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>V</mi></msup><mo>=</mo><mi>Map</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>V</mi></msup><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^V = Map(S^V,-)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is a finite-dimensional <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>V</mi></msup></mrow><annotation encoding="application/x-tex">S^V</annotation></semantics></math> is its <a class="existingWikiWord" href="/nlab/show/one-point+compactification">one-point compactification</a>. Now if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-space that can be <a class="existingWikiWord" href="/nlab/show/delooping">delooped</a> “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> times,” we can define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>V</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>V</mi></mrow></msup><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^V(X;A) = \pi_0(Map(X,\Omega^{-V} A)</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> can be delooped <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> times for all representations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, then our integer-graded cohomology theory can be expanded to an <strong><a class="existingWikiWord" href="/nlab/show/RO%28G%29-grading">RO(G)-graded</a> cohomology theory</strong>, with cohomology groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>α</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\alpha(X;A)</annotation></semantics></math> for all formal differences of representations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>=</mo><mi>V</mi><mo>−</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">\alpha = V - W</annotation></semantics></math>. The corresponding notion of spectrum is a <strong><a class="existingWikiWord" href="/nlab/show/genuine+G-spectrum">genuine G-spectrum</a></strong>, which consists of spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>V</mi></msub></mrow><annotation encoding="application/x-tex">E_V</annotation></semantics></math> for all representations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>V</mi></msub><mo>≃</mo><msup><mi>Ω</mi> <mrow><mi>W</mi><mo>−</mo><mi>V</mi></mrow></msup><msub><mi>E</mi> <mi>W</mi></msub></mrow><annotation encoding="application/x-tex">E_V \simeq \Omega^{W-V} E_W</annotation></semantics></math>. A naive Eilenberg-Mac Lane <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>-spectrum can be extended to a genuine one precisely when the coefficient system it came from can be extended to a <a class="existingWikiWord" href="/nlab/show/Mackey+functor">Mackey functor</a>, and in this case we get an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RO</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RO(G)</annotation></semantics></math>-graded Bredon cohomology theory</strong> .</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RO</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RO(G)</annotation></semantics></math>-graded Bredon cohomology has lots of formal advantages over the integer-graded theory. For instance, the torsion that popped up in odd places before can now be seen as arising by “shifting” of something in the cohomology of a point in an “off-integer dimension,” which was invisible to the integer-graded theory. Also there is a <a class="existingWikiWord" href="/nlab/show/Poincare+duality">Poincare duality</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-manifolds: if <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> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-manifold, then we can embed it in a representation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> (generally not a trivial one!) and by <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> arguments, obtain a Poincare duality theorem involving a dimension shift of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> is generally not an integer (and, apparently, not even uniquely determined by <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>!). Unfortunately, however, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RO</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RO(G)</annotation></semantics></math>-graded Bredon cohomology is kind of hard to compute.</p> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a></em> and <em><a class="existingWikiWord" href="/nlab/show/global+equivariant+stable+homotopy+theory">global equivariant stable homotopy theory</a></em>.</p> <h3 id="examples_2">Examples</h3> <ul> <li> <p><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>-equivariant cohomology theories: <a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a>, <a class="existingWikiWord" href="/nlab/show/MR-theory">MR-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+group">modular group</a>-equivariance: <a class="existingWikiWord" href="/nlab/show/modular+equivariant+elliptic+cohomology">modular equivariant elliptic cohomology</a></p> </li> </ul> <h2 id="multiplicative_equivariant_cohomology">Multiplicative equivariant cohomology</h2> <p>For <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theories">multiplicative cohomology theories</a> there is a further refinement of equivariance where the equivariant cohomology groups are built from global sections on a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> over cerain systems of <a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a>. For more on this see at</p> <ul> <li> <p><a href="http://ncatlab.org/nlab/show/A+Survey+of+Elliptic+Cohomology+-+A-equivariant+cohomology#equivariant">Equivariant multiplicative cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a></p> </li> </ul> <h2 id="examples_3">Examples</h2> <div> <table><thead><tr><th>(<a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a>) <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">representing</a> <br /> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></th><th><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> <br /> of the <a class="existingWikiWord" href="/nlab/show/point">point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">cohomology</a> <br /> of <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+ordinary+cohomology">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">HZ</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Borel+equivariant+cohomology">Borel equivariance</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>G</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KU">KU</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>KU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>R</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">KU_G(\ast) \simeq R_{\mathbb{C}}(G)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah-Segal+completion+theorem">Atiyah-Segal completion theorem</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>KU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><msub><mi>KU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>≃</mo><mi>KU</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+complex+cobordism+cohomology+theory">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/MU">MU</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MU_G(\ast)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/completion+theorem+for+complex+cobordism+cohomology">completion theorem for complex cobordism cohomology</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>MU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><msub><mi>MU</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mo>≃</mo><mi>MU</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+algebraic+K-theory">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">K \mathbb{F}_p</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub><msub><mo stretchy="false">)</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>R</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Rector+completion+theorem">Rector completion theorem</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mi>p</mi></msub><msub><mo stretchy="false">)</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><mo stretchy="false">(</mo><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub><msub><mo stretchy="false">)</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a></mtext></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi>K</mi><msub><mi>𝔽</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G) </annotation></semantics></math> <br /></td></tr> <tr><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/equivariant+stable+cohomotopy">equivariant</a>) <br /> <a class="existingWikiWord" href="/nlab/show/stable+cohomotopy">stable cohomotopy</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><msub><mi>𝔽</mi> <mn>1</mn></msub><mover><mo>≃</mo><mtext><a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a></mtext></mover></mrow><annotation encoding="application/x-tex">K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/equivariant+sphere+spectrum">S</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Burnside+ring">Burnside ring</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕊</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{S}_G(\ast) \simeq A(G)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Segal-Carlsson+completion+theorem">Segal-Carlsson completion theorem</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a></mtext></mover><msub><mi>𝕊</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mtext>compl.</mtext></mover><mover><mrow><msub><mi>𝕊</mi> <mi>G</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><mo>^</mo></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a></mtext></mover><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi>𝕊</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G) </annotation></semantics></math> <br /></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariance">equivariance</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+structure">equivariant structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+bundle">equivariant bundle</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+connection">equivariant connection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+differential+topology">equivariant differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+stable+homotopy+theory">equivariant stable homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/global+equivariant+stable+homotopy+theory">global equivariant stable homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+rational+homotopy+theory">equivariant rational homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/rational+equivariant+stable+homotopy+theory">rational equivariant stable homotopy theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cobordism+theory">equivariant cobordism theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+localization">equivariant localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Segal+conjecture">Segal conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+operator+K-theory">equivariant operator K-theory</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+KK-theory">equivariant KK-theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Baum-Connes+conjecture">Baum-Connes conjecture</a>, <a class="existingWikiWord" href="/nlab/show/Green-Julg+theorem">Green-Julg theorem</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah-Segal+completion+theorem">Atiyah-Segal completion theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+commutes+with+reduction">quantization commutes with reduction</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+complex+oriented+cohomology+theory">equivariant complex oriented cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">orbifold cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chen-Ruan+cohomology">Chen-Ruan cohomology</a>, <a class="existingWikiWord" href="/nlab/show/delocalized+equivariant+cohomology">delocalized equivariant cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+equivariant+cohomology">twisted equivariant cohomology</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> and <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> in terms of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong> (<a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">FSS 12 I, exmp. 4.4</a>):</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></th><th><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pointed+type">pointed</a> <a class="existingWikiWord" href="/nlab/show/connected+homotopy+type">connected</a> <a class="existingWikiWord" href="/nlab/show/context">context</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinvariants">coinvariants</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trivial+representation">trivial representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+invariants">homotopy invariants</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+cohomology">∞-group cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> of <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/restricted+representation">restricted representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinduced+representation">coinduced representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a> in <a class="existingWikiWord" href="/nlab/show/context">context</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum+with+G-action">spectrum with G-action</a> (<a class="existingWikiWord" href="/nlab/show/naive+G-spectrum">naive G-spectrum</a>)</td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Introduction to <a class="existingWikiWord" href="/nlab/show/Borel+equivariant+cohomology">Borel equivariant cohomology</a>:</p> <ul> <li id="Tu11"> <p><a class="existingWikiWord" href="/nlab/show/Loring+Tu">Loring Tu</a>, <em>What is… Equivariant Cohomology?</em>, Notices of the AMS, Volume 85, Number 3, March 2011 (<a href="https://www.ams.org/notices/201103/rtx110300423p.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/TuEquivariantCohomology.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Loring+Tu">Loring Tu</a>, <em>Introductory Lectures on Equivariant Cohomology</em>, Annals of Mathematics Studies <strong>204</strong>, AMS 2020 (<a href="https://press.princeton.edu/books/hardcover/9780691191744/introductory-lectures-on-equivariant-cohomology">ISBN:9780691191744</a>)</p> </li> </ul> <p>Introduction to <a class="existingWikiWord" href="/nlab/show/Bredon+equivariant+cohomology">Bredon equivariant cohomology</a>:</p> <ul> <li id="Blumberg17"> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, section 1.4 of <em>Equivariant homotopy theory</em>, 2017 (<a href="https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf">pdf</a>, <a href="https://github.com/adebray/equivariant_homotopy_theory">GitHub</a>)</p> </li> <li id="GreenleesMay"> <p><a class="existingWikiWord" href="/nlab/show/John+Greenlees">John Greenlees</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, section 3 of <em>Equivariant stable homotopy theory</em> (<a href="http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf">pdf</a>)</p> </li> <li id="Guillou06"> <p><a class="existingWikiWord" href="/nlab/show/Bert+Guillou">Bert Guillou</a>, <em>Equivariant Homotopy and Cohomology</em>, lecture notes, 2020 (<a href="http://www.ms.uky.edu/~guillou/F20/751Notes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/GuillouEquivariantHomotopyAndCohomology.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Textbooks and lecture notes:</p> <ul> <li id="tomDieck79"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, section 7 of <em><a class="existingWikiWord" href="/nlab/show/Transformation+Groups+and+Representation+Theory">Transformation Groups and Representation Theory</a></em>, Lecture Notes in Mathematics 766, Springer 1979</p> </li> <li id="May96"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a> et al., <em>Equivariant homotopy and cohomology theory</em>, CBMS Regional Conference Series in Mathematics Volume: 91; 1996 (<a href="https://bookstore.ams.org/cbms-91">ISBN:978-0-8218-0319-6</a>, <a href="http://www.math.uchicago.edu/~may/BOOKS/alaska.pdf">pdf</a>, <a href="https://ncatlab.org/nlab/files/MayEtAlEquivariant96.pdf">pdf</a>)</p> </li> <li> <p>Matvei Libine, <em>Lecture Notes on Equivariant Cohomology</em> (<a href="http://arxiv.org/abs/0709.3615">arXiv</a>)</p> </li> <li> <p>Sébastien Racanière, <em>Lecture on Equivariant Cohomology</em>, 2004 (<a class="existingWikiWord" href="/nlab/files/RacaniereEquivariant04.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>For a brief modern survey see also the first three sections of</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hill">Michael Hill</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em>The Arf-Kervaire problem in algebraic topology: Sketch of the proof</em> (<a class="existingWikiWord" href="/nlab/files/HHRKervaire.pdf" title="pdf">pdf</a>)</p> <p>(with an eye towards application to the <a class="existingWikiWord" href="/nlab/show/Arf-Kervaire+invariant+problem">Arf-Kervaire invariant problem</a>)</p> </li> <li> <p>blog on <a href="http://www.aimath.org/wiki/localization/index.php/Main_Page">Localization techniques in Equivariant Cohomology</a></p> </li> </ul> <p>Discussion of equivariant versions of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> is in</p> <ul> <li>Andreas Kübel, <a class="existingWikiWord" href="/nlab/show/Andreas+Thom">Andreas Thom</a>, <em>Equivariant Differential Cohomology</em>, Transactions of the American Mathematical Society (2018) (<a href="https://arxiv.org/abs/1510.06392">arXiv:1510.06392</a>)</li> </ul> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/equivariant+de+Rham+cohomology">equivariant de Rham cohomology</a></em>.</p> <h3 id="InComplexOrientedGeneralizedCohomologyTheory">In complex oriented generalized cohomology theory</h3> <p>Equivariant <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a> is discussed in the following articles.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael+Hopkins">Michael Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Nicholas+Kuhn">Nicholas Kuhn</a>, <a class="existingWikiWord" href="/nlab/show/Douglas+Ravenel">Douglas Ravenel</a>, <em>Generalized group characters and complex oriented cohomology theories</em>, J. Amer. Math. Soc. 13 (2000), 553-594 (<a href="http://www.ams.org/journals/jams/2000-13-03/S0894-0347-00-00332-5/">publisher</a>, <a href="http://www.math.rochester.edu/people/faculty/doug/mypapers/hkr.pdf">pdf</a>)</p> <p>(This deals with “naive” Borel-equivariant complex oriented cohomology, but discusses general <a class="existingWikiWord" href="/nlab/show/character">character</a> expressions and explicit formulas for equivariant <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">K(n)</a>-cohomology.)</p> </li> </ul> <p>Specifically equivariant <a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a> is discussed in</p> <ul> <li id="tomDiek70"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <em>Bordism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-manifolds and integrability theorems, Topology 9 (1970) 345-358</em></p> </li> <li id="Abrams13a"> <p><a class="existingWikiWord" href="/nlab/show/William+Abram">William Abram</a>, <em>Equivariant complex cobordism</em>, 2013 (<a href="http://deepblue.lib.umich.edu/handle/2027.42/99796">web</a>, <a href="http://deepblue.lib.umich.edu/bitstream/handle/2027.42/99796/abramwc_1.pdf?sequence=1">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Abram">William Abram</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <em>The equivariant complex cobordism ring of a finite abelian group</em> (<a href="http://www.math.lsa.umich.edu/~ikriz/cobordism13022-1.pdf">pdf</a>)</p> </li> </ul> <p>The following articles discuss <a class="existingWikiWord" href="/nlab/show/equivariant+formal+group+laws">equivariant formal group laws</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Greenlees">John Greenlees</a>, <em>Equivariant formal group laws and complex oriented cohomology theories</em>, Homology Homotopy Appl. Volume 3, Number 2 (2001), ii-451 (<a href="http://projecteuclid.org/euclid.hha/1139840255">EUCLID</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Abram">William Abram</a>, <em>On the equivariant formal group law of the equivariant complex cobordism ring</em>, (<a href="http://arxiv.org/abs/1309.0722">arXiv:1309.0722</a>)</p> <p>(also <a href="#Abrams13a">Abrams 13a, section III</a>).</p> </li> </ul> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/equivariant+elliptic+cohomology">equivariant elliptic cohomology</a></em>.</p> <div> <h3 id="traditional_orbifold_cohomology">Traditional orbifold cohomology</h3> <p>Traditionally, the <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> has, by and large, been taken to be simply the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of (the plain <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of) the <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">geometric realization</a> of the <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological</a>/<a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a> corresponding to the orbifold.</p> <p>For the <a class="existingWikiWord" href="/nlab/show/global+quotient+orbifold">global quotient orbifold</a> of a <a class="existingWikiWord" href="/nlab/show/topological+G-space">G-space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, this is the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of (the bare <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of) the <a class="existingWikiWord" href="/nlab/show/Borel+construction">Borel construction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>X</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>E</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">X \!\sslash\! G \;\simeq\; X \times_G E G </annotation></semantics></math>, hence is <em><a class="existingWikiWord" href="/nlab/show/Borel+cohomology">Borel cohomology</a></em> (as opposed to finer versions of <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> such as <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a>).</p> <p>A dedicated account of this Borel cohomology of orbifolds, in the generality of <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> (i.e. with <a class="existingWikiWord" href="/nlab/show/local+coefficients">local coefficients</a>) is in:</p> <ul> <li id="MoerdijkPronk99"><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Dorette+Pronk">Dorette Pronk</a>, <em>Simplicial cohomology of orbifolds</em>, Indagationes Mathematicae Volume 10, Issue 2, 1999, Pages 269-293 (<a href="https://doi.org/10.1016/S0019-3577(99)80021-4">doi:10.1016/S0019-3577(99)80021-4</a>)</li> </ul> <p>Moreover, since the orbifold’s <a class="existingWikiWord" href="/nlab/show/isotropy+groups">isotropy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">G_x</annotation></semantics></math> are, by definition, <a class="existingWikiWord" href="/nlab/show/finite+groups">finite groups</a>, their <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi><mo>≃</mo><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">\ast \!\sslash\! G \simeq B G</annotation></semantics></math> have purely <a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion</a> <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> in <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> degrees, and hence become indistinguishable from the <a class="existingWikiWord" href="/nlab/show/point">point</a> in <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational cohomology</a> (and more generally whenever their <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> is <a class="existingWikiWord" href="/nlab/show/unit">invertible</a> in the <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> <a class="existingWikiWord" href="/nlab/show/ring">ring</a>).</p> <p>Therefore, in the special case of <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational</a>/<a class="existingWikiWord" href="/nlab/show/real+cohomology">real</a>/<a class="existingWikiWord" href="/nlab/show/complex+cohomology">complex</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a>, the traditional orbifold Borel cohomology reduces further to an invariant of just (the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of) the naive quotient underlying an orbifold. For global quotient orbifolds this is the <a class="existingWikiWord" href="/nlab/show/topological+quotient+space">topological quotient space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X/G</annotation></semantics></math>.</p> <p>In this form, as an invariant of just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X/G</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/real+cohomology">real</a>/<a class="existingWikiWord" href="/nlab/show/complex+cohomology">complex</a>/<a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of orbifolds was originally introduced in</p> <ul> <li id="Satake56"><a class="existingWikiWord" href="/nlab/show/Ichiro+Satake">Ichiro Satake</a>, <em>On a generalisation of the notion of manifold</em>, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363 (<a href="https://doi.org/10.1073/pnas.42.6.359">doi:10.1073/pnas.42.6.359</a>)</li> </ul> <p>following analogous constructions in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Walter+Lewis+Baily">Walter Lewis Baily</a>, <em>On the quotient of an analytic manifold by a group of analytic homeomorphisms</em>, PNAS 40 (9) 804-808 (1954) (<a href="https://doi.org/10.1073/pnas.40.9.804">doi:10.1073/pnas.40.9.804</a>)</li> </ul> <p>Since this traditional <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational cohomology</a> of orbifolds does, hence, not actually reflect the specific nature of orbifolds, a proposal for a finer notion of orbifold cohomology was famously introduced (motivated from orbifolds as <a class="existingWikiWord" href="/nlab/show/target+spaces">target spaces</a> in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>, hence from orbifolding of <a class="existingWikiWord" href="/nlab/show/2d+CFTs">2d CFTs</a>) in</p> <ul> <li id="ChenRuan00"><a class="existingWikiWord" href="/nlab/show/Weimin+Chen">Weimin Chen</a>, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a>, <em>A New Cohomology Theory for Orbifold</em>, Commun. Math. Phys. 248 (2004) 1-31 (<a href="https://arxiv.org/abs/math/0004129">arXiv:math/0004129</a>)</li> </ul> <p>However, <a class="existingWikiWord" href="/nlab/show/Chen-Ruan+cohomology">Chen-Ruan cohomology</a> of an orbifold <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">\mathcal{X}</annotation></semantics></math> turns out to be just Borel cohomology with rational coefficients, hence is just Satake’s coarse cohomology – but applied to the <a class="existingWikiWord" href="/nlab/show/inertia+orbifold">inertia orbifold</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">\mathcal{X}</annotation></semantics></math>. A review that makes this nicely explicit is (see p. 4 and 7):</p> <ul> <li id="Clader14">Emily Clader, <em>Orbifolds and orbifold cohomology</em>, 2014 (<a href="http://www-personal.umich.edu/~eclader/OctLect1.pdf">pdf</a>)</li> </ul> <p>Hence <a class="existingWikiWord" href="/nlab/show/Chen-Ruan+cohomology">Chen-Ruan cohomology</a> of a global quotient orbifold is equivalently the <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational cohomology</a> (<a class="existingWikiWord" href="/nlab/show/real+cohomology">real cohomology</a>, <a class="existingWikiWord" href="/nlab/show/complex+cohomology">complex cohomology</a>) for the <a class="existingWikiWord" href="/nlab/show/topological+quotient+space">topological quotient space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>AutMor</mi><mo stretchy="false">(</mo><mi>X</mi><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">AutMor(X\!\sslash\!G)/G</annotation></semantics></math> of the space of <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> in the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/conjugation">conjugation</a> <a class="existingWikiWord" href="/nlab/show/action">action</a>.</p> <p>On the other hand, it was observed in (see p. 18)</p> <ul> <li id="Moerdijk02"><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <em>Orbifolds as Groupoids: an Introduction</em>, <a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, <a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a> (eds.) <em><a class="existingWikiWord" href="/nlab/show/Orbifolds+in+Mathematics+and+Physics">Orbifolds in Mathematics and Physics</a></em>, Contemporary Math 310 , AMS (2002), 205–222 (<a href="http://arxiv.org/abs/math.DG/0203100">arXiv:math.DG/0203100</a>)</li> </ul> <p>that for global quotient orbifolds Chen-Ruan cohomology indeed is equivalent to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-equivariant <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon 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> – for one specific choice of <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a> <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> system (<a class="existingWikiWord" href="/nlab/show/abelian+sheaf">abelian sheaf</a> on the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>), namely for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>↦</mo><mi>ClassFunctions</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G/H \mapsto ClassFunctions(H)</annotation></semantics></math>.</p> <p>Or rather, <a href="#Moerdijk02">Moerdijk 02, p. 18</a> observes that the Chen-Ruan cohomology of a global quotient orbifold is equivalently the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> of the naive quotient space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X/G</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf">abelian sheaf</a> whose <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>X</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">[x] \in X/G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/ring">ring</a> of <a class="existingWikiWord" href="/nlab/show/class+functions">class functions</a> of the <a class="existingWikiWord" href="/nlab/show/isotropy+group">isotropy group</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>; and then appeals to Theorem 5.5 in</p> <ul> <li id="Honkasalu90"><a class="existingWikiWord" href="/nlab/show/Hannu+Honkasalo">Hannu Honkasalo</a>, <em>Equivariant Alexander-Spanier cohomology for actions of compact Lie groups</em>, Mathematica Scandinavica Vol. 67, No. 1 (1990), pp. 23-34 (<a href="https://www.jstor.org/stable/24492569">jstor:24492569</a>)</li> </ul> <p>for the followup statement that the <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X/G</annotation></semantics></math> with coefficient sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>A</mi><mo>̲</mo></munder></mrow><annotation encoding="application/x-tex">\underline{A}</annotation></semantics></math> being “<a class="existingWikiWord" href="/nlab/show/locally+constant+sheaf">locally constant</a> except for dependence on <a class="existingWikiWord" href="/nlab/show/isotropy+groups">isotropy groups</a>” is equivalently <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon 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> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>↦</mo><msub><munder><mi>A</mi><mo>̲</mo></munder> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">G/H \mapsto \underline{A}_x</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Isotr</mi> <mi>x</mi></msub><mo>=</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">Isotr_x = H</annotation></semantics></math>. This identification of the coefficient systems is Prop. 6.5 b) in:</p> <ul> <li id="Honkasalo88"><a class="existingWikiWord" href="/nlab/show/Hannu+Honkasalo">Hannu Honkasalo</a>, <em>Equivariant Alexander-Spanier cohomology</em>, Mathematica Scandinavia <strong>63</strong> (1988) 179-195 [<a href="https://doi.org/10.7146/math.scand.a-12232">doi:10.7146/math.scand.a-12232</a>]</li> </ul> <p>See also Section 4.3 of</p> <ul> <li id="SzaboValentino07"><a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <a class="existingWikiWord" href="/nlab/show/Alessandro+Valentino">Alessandro Valentino</a>, <em>Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory</em>, Commun. Math. Phys. <strong>294</strong> (2010) 647-702 [<a href="https://doi.org/10.1007/s00220-009-0975-1">doi:10.1007/s00220-009-0975-1</a>, <a href="https://arxiv.org/abs/0710.2773">arXiv:0710.2773</a>]</li> </ul> <p>In summary:</p> <ul> <li> <p>Traditional orbifold cohomology theory is <a class="existingWikiWord" href="/nlab/show/Borel+cohomology">Borel cohomology</a> of underlying <a class="existingWikiWord" href="/nlab/show/Borel+construction">Borel construction</a>-spaces, and reduces <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rationally</a> further to the <a class="existingWikiWord" href="/nlab/show/rational+cohomology">rational cohomology</a> of underlying naive <a class="existingWikiWord" href="/nlab/show/quotient+spaces">quotient spaces</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chen-Ruan+cohomology">Chen-Ruan cohomology</a> is just the latter rational cohomology but applied after passage to the <a class="existingWikiWord" href="/nlab/show/inertia+orbifold">inertia orbifold</a>. This is equivalent to the <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a> of the original orbifold, for one specific <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant coefficient</a>-system.</p> </li> </ul> <p>This suggests, of course, that more of <a class="existingWikiWord" href="/nlab/show/proper+equivariant+homotopy+theory">proper</a> <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> should be brought to bear on a theory of orbifold cohomology. A partial way to achieve this is to prove for a given <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a>-theory that it descends from an invariant of <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</a> to one of the associated <a class="existingWikiWord" href="/nlab/show/global+quotient+orbifolds">global quotient orbifolds</a>.</p> <p>For <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological</a> <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a> this is the case, by</p> <ul> <li id="PronkScull07"><a class="existingWikiWord" href="/nlab/show/Dorette+Pronk">Dorette Pronk</a>, <a class="existingWikiWord" href="/nlab/show/Laura+Scull">Laura Scull</a>, Prop. 4.1 in: <em>Translation Groupoids and Orbifold Bredon Cohomology</em>, Canad. J. Math. 62(2010), 614-645 (<a href="https://arxiv.org/abs/0705.3249">arXiv:0705.3249</a>, <a href="https://doi.org/10.4153/CJM-2010-024-1">doi:10.4153/CJM-2010-024-1</a>)</li> </ul> <p>Therefore it makes sense to <em>define</em> <a class="existingWikiWord" href="/nlab/show/orbifold+K-theory">orbifold K-theory</a> for <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi></mrow><annotation encoding="application/x-tex">\mathcal{X}</annotation></semantics></math> which are equivalent to a <a class="existingWikiWord" href="/nlab/show/global+quotient+orbifold">global quotient orbifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi><mo>≃</mo><mo>≺</mo><mo stretchy="false">(</mo><mi>X</mi><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{X} \simeq \prec(X \!\sslash\! G) </annotation></semantics></math> to be the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a> of <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>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>𝒳</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msubsup><mi>K</mi> <mi>G</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K^\bullet(\mathcal{X}) \;\coloneqq\; K_G^\bullet(X) \,. </annotation></semantics></math></p> <p>This is the approach taken in</p> <ul> <li id="AdemRuan01"><a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a>, Section 3 of: <em>Twisted Orbifold K-Theory</em>, Commun. Math. Phys. 237 (2003) 533-556 (<a href="https://arxiv.org/abs/math/0107168">arXiv:math/0107168</a>)</li> </ul> <p>Exposition and review of traditional orbifold cohomology, with an emphasis on <a class="existingWikiWord" href="/nlab/show/Chen-Ruan+cohomology">Chen-Ruan cohomology</a> and <a class="existingWikiWord" href="/nlab/show/orbifold+K-theory">orbifold K-theory</a>, is in:</p> <ul> <li id="ALR07"><a class="existingWikiWord" href="/nlab/show/Alejandro+Adem">Alejandro Adem</a>, <a class="existingWikiWord" href="/nlab/show/Johann+Leida">Johann Leida</a>, <a class="existingWikiWord" href="/nlab/show/Yongbin+Ruan">Yongbin Ruan</a>, <em>Orbifolds and Stringy Topology</em>, Cambridge Tracts in Mathematics <strong>171</strong> (2007) (<a href="https://doi.org/10.1017/CBO9780511543081">doi:10.1017/CBO9780511543081</a>, <a href="http://www.math.colostate.edu/~renzo/teaching/Orbifolds/Ruan.pdf">pdf</a>)</li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on July 30, 2024 at 15:14:51. 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