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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Bredon cohomology </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/652/#Item_34" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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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/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="topos_theory"><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 Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></strong></p> <h2 id="sidebar_background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elementary+%28%E2%88%9E%2C1%29-topos">elementary (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization of an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-category of (∞,1)-sheaves</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>/<a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-topos">(n,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/n-topos">n-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/truncated">n-truncated object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected">n-connected object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">(1,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-presheaf">(2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-quasitopos">(∞,1)-quasitopos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+%28%E2%88%9E%2C1%29-presheaf">separated (∞,1)-presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+presheaf">separated presheaf</a></li> </ul> </li> <li> <p><span class="newWikiWord">(2,1)-quasitopos<a href="/nlab/new/%282%2C1%29-quasitopos">?</a></span></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/separated+%282%2C1%29-presheaf">separated (2,1)-presheaf</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-topos">(∞,2)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-topos">(∞,n)-topos</a></p> </li> </ul> <h2 id="sidebar_characterization">Characterization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+colimits">universal colimits</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a></li> </ul> </li> </ul> <h2 id="sidebar_morphisms">Morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Topos">(∞,1)Topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lawvere+distribution">Lawvere distribution</a></p> </li> </ul> <h2 id="sidebar_extra">Extra stuff, structure and property</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+%28%E2%88%9E%2C1%29-topos">hypercomplete (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercomplete+object">hypercomplete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+theorem">Whitehead theorem</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over-%28%E2%88%9E%2C1%29-topos">over-(∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-localic+%28%E2%88%9E%2C1%29-topos">n-localic (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+n-connected+%28n%2C1%29-topos">locally n-connected (n,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <h2 id="sidebar_models">Models</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/models+for+%E2%88%9E-stack+%28%E2%88%9E%2C1%29-toposes">models for ∞-stack (∞,1)-toposes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+functors">model structure on functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+site">model site</a>/<a class="existingWikiWord" href="/nlab/show/sSet-site">sSet-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Verity+on+descent+for+strict+omega-groupoid+valued+presheaves">descent for presheaves with values in strict ∞-groupoids</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_constructions">Constructions</h2> <p><strong>structures in a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+of+an+%28%E2%88%9E%2C1%29-topos">shape</a> / <a class="existingWikiWord" href="/nlab/show/coshape+of+an+%28%E2%88%9E%2C1%29-topos">coshape</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a>/<a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical</a>/<a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric</a> homotopy groups</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+tower+in+an+%28%E2%88%9E%2C1%29-category">Postnikov tower</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead+tower+in+an+%28%E2%88%9E%2C1%29-topos">Whitehead tower</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory+in+an+%28%E2%88%9E%2C1%29-topos">rational homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+dimension">homotopy dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomological+dimension">cohomological dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covering+dimension">covering dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+dimension">Heyting dimension</a></p> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/%28infinity%2C1%29-topos+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#with_coefficients_in_representation_ring'>With coefficients in representation ring</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>What is called <em>Bredon cohomology</em> after (<a href="#Bredon67a">Bredon 67a</a>, <a href="#Bredon67a">Bredon 67a</a>) is the flavor of <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> which uses the “fine” <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</a> that by <a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a> is equivalent to the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaves">(∞,1)-presheaves</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a>, instead of the “coarse” <a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel homotopy theory</a>. See at <em><a href="equivariant+cohomology#Idea">Equivariant cohomology – Idea</a></em> for more motivation.</p> <p>For more technical details see there <em><a href="equivariant+cohomology#Bredon">equivariant cohomology – Bredon equivariant cohomology</a></em>.</p> <h2 id="Definition">Definition</h2> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Orb</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">Orb_G</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a> and write <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><msub><mi>Orb</mi> <mi>G</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_\infty(Orb_G)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Orb</mi> <mi>G</mi></msub></mrow><annotation encoding="application/x-tex">Orb_G</annotation></semantics></math>. By <a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a> this is <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</a> with weak equivalences 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>-<a class="existingWikiWord" href="/nlab/show/fixed+point">fixed point</a>-wise <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> for all closed subgroups <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> (“the <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a>”):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup></mrow></msup><mo>≔</mo><msub><mi>L</mi> <mi>fpwe</mi></msub><mi>G</mi><mi>Top</mi><mo>≃</mo><msub><mi>PSh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><msub><mi>Orb</mi> <mi>G</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}^{Orb_G^{op}} \coloneqq L_{fpwe} G Top \simeq PSh_\infty(Orb_G) \,. </annotation></semantics></math></div> <p>A <a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><mi>Stab</mi><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \in Stab(\mathbf{H}^{Orb_G^{op}})</annotation></semantics></math> 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><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{Orb_G^{op}}</annotation></semantics></math> is what is called a <a class="existingWikiWord" href="/nlab/show/spectrum+with+G-action">spectrum with G-action</a> or, for better or worse, a “<a class="existingWikiWord" href="/nlab/show/naive+G-spectrum">naive G-spectrum</a>”.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/G-space">G-space</a>, then its <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{Orb_G^{op}}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in such <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> might be called <em>generalized Bredon cohomology</em> (in the “generalized” sense of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a>).</p> <p>Specifically for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>Ab</mi><mo stretchy="false">(</mo><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>Orb</mi> <mi>G</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in Ab(Sh(Orb_G))</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/abelian+sheaf">abelian sheaf</a> then there is an <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+object">Eilenberg-MacLane object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup></mrow></msup></mrow><annotation encoding="application/x-tex"> K(n,A) \in \mathbf{H}^{Orb_G^{op}} </annotation></semantics></math></div> <p>whose <a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos">categorical homotopy groups</a> are concentrated in degree <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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>Then <em>ordinary Bredon cohomology</em> (in the “ordinary” sense of <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>) in degree <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> with <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><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is cohomology in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{Orb_G^{op}}</annotation></semantics></math> with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(n,A)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>G</mi> <mi>n</mi></msubsup><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><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H_G^n(X,A) \simeq \pi_0 \mathbf{H}^{Orb_G^{op}}(X,A) </annotation></semantics></math></div> <p>(see the general discussion at <em><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em>).</p> <p>If here <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 presented by a <a class="existingWikiWord" href="/nlab/show/G-CW+complex">G-CW complex</a> and hence is cofibrant in the <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure that presents the <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a> (see at <em><a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a></em> for details), then the <a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a> on the right above is equivalently given by 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 ordinary <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> of the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> underlying the <a class="existingWikiWord" href="/nlab/show/G-spaces">G-spaces</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>G</mi> <mi>n</mi></msubsup><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>X</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mi>G</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_G^n(X,A) \simeq \pi_0 [X,A]_G \,. </annotation></semantics></math></div> <p>In this form ordinary Bredon cohomology is expressed in (<a href="#Bredon67a">Bredon 67a, p. 3</a>, <a href="#Bredon67b">Bredon 67b, Theorem (2.11), (6.1)</a>), review in in (<a href="#GreenleesMay">Greenlees-May, p. 10</a>).</p> <p>The definition of Bredon cohomology which is more popular (<a href="#Bredon67a">Bredon 67a, p. 2</a>, <a href="#Bredon67b">Bredon 67b, I.6</a>) is a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>-model for this: regarding <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> again as a presheaf on the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a>, define a presheaf of chain complexes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Orb</mi> <mi>G</mi> <mi>op</mi></msubsup><mo>⟶</mo><msub><mi>Ch</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> C_\bullet(X) \;\colon\; Orb_G^{op}\longrightarrow Ch_\bullet </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>X</mi> <mi>n</mi></msup><msup><mo stretchy="false">)</mo> <mi>H</mi></msup><mo>,</mo><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo> <mi>H</mi></msup><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> C_n(X)(G/H) \coloneqq H_n((X^n)^H, (X^{n-1})^H, \mathbb{Z}) \,, </annotation></semantics></math></div> <p>where on the right we have the <a class="existingWikiWord" href="/nlab/show/relative+homology">relative homology</a> of the <a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a> decomposition underlying the <a class="existingWikiWord" href="/nlab/show/G-CW+complex">G-CW complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in degrees as indicated. The <a class="existingWikiWord" href="/nlab/show/differential">differential</a> on these chain complexes is defined in the obvious way (…).</p> <p>Then one has an expression for <em>ordinary Bredon cohomology</em> similar to that of <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>H</mi> <mi>G</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>Hom</mi> <mrow><msub><mi>Orb</mi> <mi>G</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>C</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_G^n(X,A) \simeq H_n(Hom_{Orb_G}(C_\bullet(X), A)) \,. </annotation></semantics></math></div> <p>(due to <a href="#Bredon67">Bredon 67</a>, see e.g. (<a href="#GreenleesMay">Greenlees-May, p. 9</a>)).</p> <p>More generally there is <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 <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <a class="existingWikiWord" href="/nlab/show/genuine+G-spectra">genuine G-spectra</a>. This is <em>also</em> sometimes still referred to as “Bredon cohomology”. For more on this see at <em><a href="equivariant%20cohomology#Bredon">equivariant cohomology – Bredon cohonology</a></em>.</p> <h2 id="examples">Examples</h2> <h3 id="with_coefficients_in_representation_ring">With coefficients in representation ring</h3> <div> <p><strong>Incarnations of <a class="existingWikiWord" href="/nlab/show/rational+equivariant+K-theory">rational equivariant K-theory</a>:</strong></p> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> theory</th><th>definition/equivalence due to</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msubsup><mi>K</mi> <mi>G</mi> <mn>0</mn></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>;</mo><mi>ℂ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\simeq K_G^0\big(X; \mathbb{C} \big) </annotation></semantics></math></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/rational+equivariant+K-theory">rational equivariant K-theory</a></strong></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msup><mi>H</mi> <mi>ev</mi></msup><mo maxsize="1.8em" minsize="1.8em">(</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><msup><mi>X</mi> <mi>g</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mo stretchy="false">/</mo><mi>G</mi><mo>;</mo><mi>ℂ</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> \simeq H^{ev}\Big( \big(\underset{g \in G}{\coprod} X^g\big)/G; \mathbb{C} \Big) </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/delocalized+equivariant+cohomology">delocalized equivariant cohomology</a></td><td style="text-align: left;"><a href="delocalized+equivariant+cohomology#ConnesBaum89">Baum-Connes 89, Thm. 1.19</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msubsup><mi>H</mi> <mi>CR</mi> <mi>ev</mi></msubsup><mo maxsize="1.8em" minsize="1.8em">(</mo><mo>≺</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex">\simeq H^{ev}_{CR}\Big( \prec \big( X \!\sslash\! G\big);\, \mathbb{C} \Big)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Chen-Ruan+cohomology">Chen-Ruan cohomology</a> <br /> of <a class="existingWikiWord" href="/nlab/show/global+quotient+orbifold">global quotient orbifold</a></td><td style="text-align: left;"><a href="Chen-Ruan+cohomology#ChenRuan00">Chen-Ruan 00, Sec. 3.1</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msubsup><mi>H</mi> <mi>G</mi> <mi>ev</mi></msubsup><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>X</mi><mo>;</mo><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>↦</mo><mi>ℂ</mi><mo>⊗</mo><mi>Rep</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex">\simeq H^{ev}_G\Big( X; \, \big(G/H \mapsto \mathbb{C} \otimes Rep(H)\big) \Big)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a> <br /> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <a class="existingWikiWord" href="/nlab/show/representation+ring">representation ring</a></td><td style="text-align: left;"><a href="orbifold+cohomology#Honkasalo88">Ho88 6.5</a>+<a href="orbifold+cohomology#Honkasalu90">Ho90 5.5</a>+<a href="orbifold+cohomology#Moerdijk02">Mo02 p. 18</a>, <br /> <a href="equivariant+Chern+character#MislinValette03">Mislin-Valette 03, Thm. 6.1</a>, <br /> <a href="orbifold+cohomology#SzaboValentino07">Szabo-Valentino 07, Sec. 4.2</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo><msubsup><mi>K</mi> <mi>G</mi> <mn>0</mn></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>;</mo><mi>ℂ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\simeq K_G^0\big(X; \mathbb{C} \big) </annotation></semantics></math></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/rational+equivariant+K-theory">rational equivariant K-theory</a></strong></td><td style="text-align: left;"><a href="equivariant+K-theory#LueckOliver01">Lück-Oliver 01, Thm. 5.5</a>, <br /> <a href="equivariant+Chern+character#MislinValette03">Mislin-Valette 03, Thm. 6.1</a></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Elmendorf%27s+theorem">Elmendorf's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+Bredon+cohomology">twisted Bredon cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+equivariant+homotopy+theory">global equivariant homotopy theory</a></p> </li> <li> <p><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/orbifold+cohomology">orbifold cohomology</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in the presence of <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> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Borel+equivariant+cohomology">Borel equivariant cohomology</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AAA</mi></mphantom><mo>←</mo><mphantom><mi>AAA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AAA}\leftarrow\phantom{AAA}</annotation></semantics></math></th><th>general (<a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon</a>) <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AAA</mi></mphantom><mo>→</mo><mphantom><mi>AAA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AAA}\rightarrow\phantom{AAA}</annotation></semantics></math></th><th>non-equivariant <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> with <a class="existingWikiWord" href="/nlab/show/homotopy+fixed+point">homotopy fixed point</a> <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>G</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}</annotation></semantics></math></td><td style="text-align: left;">trivial <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">action</a> on <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</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><mphantom><mi>AA</mi></mphantom><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><msup><mo stretchy="false">]</mo> <mi>G</mi></msup><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}[X,A]^G\phantom{AA}</annotation></semantics></math></td><td style="text-align: left;">trivial <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">action</a> on <a class="existingWikiWord" href="/nlab/show/domain">domain</a> space <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></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>A</mi> <mi>G</mi></msup><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}\mathbf{H}(X, A^G)\phantom{AA}</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>The original text:</p> <ul> <li id="Bredon67b"><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, <em><a class="existingWikiWord" href="/nlab/show/Equivariant+cohomology+theories">Equivariant cohomology theories</a></em>, Springer Lecture Notes in Mathematics Vol. 34. 1967 (<a href="https://link.springer.com/book/10.1007/BFb0082690">doi:10.1007/BFb0082690</a>)</li> </ul> <p>announced in</p> <ul> <li id="Bredon67a"><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, <em>Equivariant cohomology theories</em>, Bull. Amer. Math. Soc. Volume 73, Number 2 (1967), 266-268. (<a href="https://projecteuclid.org/euclid.bams/1183528794">educlid:1183528794</a>)</li> </ul> <p>Also:</p> <ul> <li id="Illman72"> <p><a class="existingWikiWord" href="/nlab/show/S%C3%B6ren+Illman">Sören Illman</a>, Chapter III in: <em>Equivariant Algebraic Topology</em>, Princeton University 1972 (<a class="existingWikiWord" href="/nlab/files/Illman_EquivariantAlgebraicTopology1972.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S%C3%B6ren+Illman">Sören Illman</a>, <em>Equivariant singular homology and cohomology</em>, Bull. Amer. Math. Soc. Volume 79, Number 1 (1973), 188-192 (<a href="https://projecteuclid.org/euclid.bams/1183534324">euclid:bams/1183534324</a>)</p> </li> </ul> <p>Review:</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="tomDieck87"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, Section II.9 of: <em><a class="existingWikiWord" href="/nlab/show/Transformation+Groups">Transformation Groups</a></em>, de Gruyter 1987 (<a href="https://doi.org/10.1515/9783110858372">doi:10.1515/9783110858372</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>, pages 9-10 of <em><a class="existingWikiWord" href="/nlab/show/Equivariant+stable+homotopy+theory">Equivariant stable homotopy theory</a></em>, chapter 8, pages 277-323 in: <a class="existingWikiWord" href="/nlab/show/Ioan+James">Ioan James</a> (ed.), <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em>, North-Holland, Amsterdam, 1995. (<a href="https://doi.org/10.1016/B978-0-444-81779-2.X5000-7">doi:10.1016/B978-0-444-81779-2.X5000-7</a>, <a href="http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf">pdf</a>)</p> </li> <li id="May96"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, section I.4 of <em>Equivariant homotopy and cohomology theory</em> CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comezana, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (<a href="https://web.math.rochester.edu/people/faculty/doug/otherpapers/alaska1.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MayEtAlEquivariant96.pdf" title="pdf">pdf</a>)</p> </li> <li> <p>Paolo Masulli, section 2 of <em>Equivariant homotopy: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KR</mi></mrow><annotation encoding="application/x-tex">KR</annotation></semantics></math>-theory</em>, Master thesis (2011) (<a href="http://www.math.ku.dk/~jg/students/masulli.msthesis.2011.pdf">pdf</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+objects">Eilenberg-MacLane objects</a> over the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a> are discussed in detail in</p> <ul> <li id="Lewis92"><a class="existingWikiWord" href="/nlab/show/L.+Gaunce+Lewis%2C+Jr.">L. Gaunce Lewis, Jr.</a>, <em>Equivariant Eilenberg-MacLane spaces and the equivariant Seifert-van Kampen suspension theorems</em>, Topology Appl., 48 (1992), no. 1, pp. 25–61 (<a href="https://doi.org/10.1016/0166-8641(92)90120-O">doi:10.1016/0166-8641(92)90120-O</a>)</li> </ul> <p>Equivalence of Bredon cohomology of <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</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> to <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> of 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> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> a “<a class="existingWikiWord" href="/nlab/show/locally+constant+sheaf">locally constant sheaf</a> except for dependence on <a class="existingWikiWord" href="/nlab/show/isotropy+groups">isotropy groups</a>”:</p> <ul> <li id="Honkasalo88"> <p><a class="existingWikiWord" href="/nlab/show/Hannu+Honkasalo">Hannu Honkasalo</a>, <em>Equivariant Alexander-Spanier cohomology</em>, Mathematica Scandinavia, 63, 179-195, 1988 (<a href="https://doi.org/10.7146/math.scand.a-12232">doi:10.7146/math.scand.a-12232</a>)</p> <p>(for <a class="existingWikiWord" href="/nlab/show/finite+groups">finite groups</a>)</p> </li> <li id="Honkasalu90"> <p><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>)</p> <p>(for <a class="existingWikiWord" href="/nlab/show/compact+Lie+groups">compact Lie groups</a>)</p> </li> <li id="Honkasalo96"> <p><a class="existingWikiWord" href="/nlab/show/Hannu+Honkasalo">Hannu Honkasalo</a>, <em>Sheaves on fixed point sets and equivariant cohomology</em>, Math. Scand. <strong>78</strong> (1996), 37–55 (<a href="https://www.jstor.org/stable/24492815">jstor:</a>)</p> <p>(reformulation in <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a>)</p> </li> </ul> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">orbifold cohomology</a></em>.</p> <p>Equivalent formulation using the <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> for a certain <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>-valued <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a>“</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">I. Moerdijk</a>, J-A. Svensson, <em>The equivariant Serre spectral sequence</em>, Proc. Amer. Math. Soc. <strong>118</strong> (1993), no. 1, 263–278 <a href="http://dx.doi.org/10.2307/2160037">doi:10.2307/2160037</a></li> </ul> <p>Further remarks on this and on the <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>-version is in</p> <ul> <li> <p>Goutam Mukherjee, N. Pandey, <em>Equivariant cohomology with local coefficients</em> (<a href="http://www.ams.org/proc/2002-130-01/S0002-9939-01-06377-8/S0002-9939-01-06377-8.pdf">pdf</a>)</p> </li> <li> <p>Amiya Mukherjee, Goutam Mukherjee, <em>Bredon-Illman cohomology with local coefficients</em>, The Quarterly Journal of Mathematics, Volume 47, Issue 2, June 1996, Pages 199–219 (<a href="https://doi.org/10.1093/qmath/47.2.199">doi:10.1093/qmath/47.2.199</a>)</p> </li> <li id="Honkasalo97"> <p><a class="existingWikiWord" href="/nlab/show/Hannu+Honkasalo">Hannu Honkasalo</a>, <em>A sheaf-theoretic approach to the equivariant Serre spectral sequence</em>, J. Math. Sci. Univ. Tokyo 4 (1997), 53–65 (<a href="http://journal.ms.u-tokyo.ac.jp/pdf/jms040103.pdf">pdf</a>)</p> </li> <li> <p>Samik Basua, Debasis Sen, <em>Representing Bredon cohomology with local coefficients</em>, Journal of Pure and Applied Algebra Volume 219, Issue 9, September 2015, Pages 3992-4015 (<a href="https://doi.org/10.1016/j.jpaa.2015.02.001">doi:10.1016/j.jpaa.2015.02.001</a>)</p> </li> </ul> <p>On cases where <a class="existingWikiWord" href="/nlab/show/prime+field"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>ℤ</mi> <mo stretchy="false">/</mo> <mi>p</mi> </mrow> <annotation encoding="application/x-tex">\mathbb{Z}/p</annotation> </semantics> </math></a>-<a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant</a> <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a> groups are <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a> over the Bredon cohomology of the point:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kevin+K.+Ferland">Kevin K. Ferland</a>, <a class="existingWikiWord" href="/nlab/show/L.+Gaunce+Lewis%2C+Jr.">L. Gaunce Lewis, Jr.</a>, <em>The <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 Equivariant Ordinary Homology 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>-Cell Complexes with Even-Dimensional Cells for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">G=\mathbb{Z}/p</annotation></semantics></math></em>, Memoirs of the AMS <strong>167</strong> (2004) [<a href="https://bookstore.ams.org/memo-167-794">ams:memo-167-794</a>, 978-ISBN:1-4704-0392-8]</li> </ul> <p>Specifically on <a class="existingWikiWord" href="/nlab/show/Kronholm%27s+freeness+theorem">Kronholm's freeness theorem</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-equivariant Bredon cohomology:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/William+C.+Kronholm">William C. Kronholm</a>, <em>A Freeness Theorem for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RO</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RO(\mathbb{Z}/2)</annotation></semantics></math>-graded Cohomology</em>, Topology and its Applications <strong>157</strong> 5 (2010) 902-915 [<a href="https://arxiv.org/abs/0908.3825">arXiv:0908.3825</a>, <a href="https://doi.org/10.1016/j.topol.2009.12.006">doi:10.1016/j.topol.2009.12.006</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eric+Hogle">Eric Hogle</a>, <a class="existingWikiWord" href="/nlab/show/Clover+May">Clover May</a>, <em>The freeness theorem for equivariant cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rep</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep(C_2)</annotation></semantics></math>-complexes</em>, Topology and its Applications <strong>285</strong> 1 (2020) 107413 [<a href="https://arxiv.org/abs/2005.07300">arXiv:2005.07300</a>, <a href="https://doi.org/10.1016/j.topol.2020.107413">doi:10.1016/j.topol.2020.107413</a>]</p> </li> <li id="DuggerHazelMay24"> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <a class="existingWikiWord" href="/nlab/show/Christy+Hazel">Christy Hazel</a>, <a class="existingWikiWord" href="/nlab/show/Clover+May">Clover May</a>, <em>Equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/\ell</annotation></semantics></math>-modules for the cyclic group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">C_2</annotation></semantics></math></em>, Journal of Pure and Applied Algebra <strong>228</strong> 3 (2024) 107473 [<a href="https://arxiv.org/abs/2203.05287">arXiv:2203.05287</a>, <a href="https://doi.org/10.1016/j.jpaa.2023.107473">doi:10.1016/j.jpaa.2023.107473</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 13, 2023 at 14:02:55. 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