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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="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#list_of_examples'>List of examples</a></li> <li><a href='#CupPowersInMultiplicativeCohomology'>Cup powers in multiplicative cohomology</a></li> </ul> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#on_ordinary_cohomology'>On ordinary cohomology</a></li> <li><a href='#on_topological_ktheory'>On topological K-theory</a></li> <li><a href='#on_complexoriented_cohomology_theories'>On complex-oriented cohomology theories</a></li> <li><a href='#on_differential_cohomology'>On differential cohomology</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>cohomology operation</em> is a family of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> between <a class="existingWikiWord" href="/nlab/show/cohomology+groups">cohomology groups</a>, which is <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a> with respect to the base space.</p> <p>Equivalently, if the <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a> has a <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> (as it does for all usual notions of <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, in particular for all <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology+theories">Whitehead-generalized cohomology theories</a>) then, by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, (non-abelian/unstable) cohomology operations are in natural bijection with <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of maps between classifying spaces.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>l</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>≃</mo><mi>Ho</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>k</mi></msub><mo>,</mo><msub><mi>F</mi> <mi>l</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{ H^k(-, E) \to H^l(-, F) \} \simeq Ho(E_k, F_l) \,. </annotation></semantics></math></div> <p>(This statement is made fully explicit for instance below def. 12.3.22 in (<a href="#AguilarGitlerPrieto">Aguilar-Gitler-Prieto</a>).</p> <p>If here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">E_\bullet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">F_\bullet</annotation></semantics></math> are the component spaces of <a class="existingWikiWord" href="/nlab/show/spectra">spectra</a> and if such cohomology operations are compatible with the <a class="existingWikiWord" href="/nlab/show/suspension+isomorphism">suspension isomorphism</a>, then one speaks of a <em>stable cohomology operation</em>.</p> <h2 id="examples">Examples</h2> <h3 id="list_of_examples">List of examples</h3> <ul> <li> <p>every <a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a> is a cohomology operation.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</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/Massey+product">Massey product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/power+operation">power operation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod+squares">Steenrod squares</a> are the stable cohomology endo-operations on <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> (mod 2)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Adams+operations">Adams operations</a> are the endo-cohomology operations on <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/logarithmic+cohomology+operation">logarithmic cohomology operation</a></p> </li> </ul> <h3 id="CupPowersInMultiplicativeCohomology">Cup powers in multiplicative cohomology</h3> <p> <div class='num_remark' id='CupSquareCohomologyOperation'> <h6>Example</h6> <p>[Cup-square cohomology operation]</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative</a> <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology+theory">Whitehead-generalized cohomology theory</a> <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by a <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>E</mi><mo>,</mo><msup><mn>1</mn> <mi>E</mi></msup><mo>,</mo><msup><mi>m</mi> <mi>E</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>CommutativeMonoids</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>Ho</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Spectra</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>,</mo><mi>𝕊</mi><mo>,</mo><mo>∧</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \big( E, 1^E, m^E \big) \;\in\; CommutativeMonoids \Big( Ho\big( Spectra\big), \mathbb{S}, \wedge \Big) \,. </annotation></semantics></math></div> <p>Then for all <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> there is an unstable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-cohomology operation</p> <div class="maruku-equation" id="eq:CupSquareAsUnstableCohomologyOperation"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">[</mo><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mi>n</mi></msup><mi>E</mi><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><msub><mn>2</mn> <mo>∪</mo></msub></mrow></msup></mrow></mover><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mi>E</mi><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mover><mi>E</mi><mo>˜</mo></mover> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Ω</mi> <mrow><mn>∞</mn><mo>−</mo><mi>n</mi></mrow></msup><mi>E</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \big[ \Omega^\infty \Sigma^n E \overset{ (-)^{2_\cup} }{\longrightarrow} \Omega^\infty \Sigma^{2 n} E \big] \;\;\in\;\; \widetilde E^{ 2 n } \big( \Omega^{\infty - n} E \big) </annotation></semantics></math></div> <p>defined – via the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> on the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of the <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a> of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> – by acting over any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>Ho</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>PointedTopologicalSpaces</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">X \,\in\, Ho\big(PointedTopologicalSpaces\big)</annotation></semantics></math> as the <a href="cup+product#CupProductInWhiteheadGeneralizedCohomologyTheory">cup square on E-cohomology</a> on cohomology 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>:</p> <div class="maruku-equation" id="eq:CupSquareInMultiplicativeCohomologyNaturalTransformation"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">(</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mi>n</mi></msup><mi>E</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Δ</mi> <mrow><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mo>∪</mo> <mi>X</mi> <mi>E</mi></msubsup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mi>E</mi><mo stretchy="false">]</mo><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> X \;\;\mapsto\;\; \Big( [X, \Omega^\infty \Sigma^n E] \,=\, {\widetilde E}{}^n(X) \overset{ \;\; \Delta_{{\widetilde E}{}^n(X)} \;\; }{\longrightarrow} {\widetilde E}{}^n(X) \times {\widetilde E}{}^n(X) \overset{ \;\; \cup^E_X \;\; }{\longrightarrow} {\widetilde E}{}^n(X) \,=\, [X, \Omega^\infty \Sigma^{2n} E] \Big) </annotation></semantics></math></div> <p>Here the first <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><mover><mi>E</mi><mo>˜</mo></mover><msup><mrow></mrow> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\Delta_{{\widetilde E}{}^n(X)}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/diagonal+morphism">diagonal</a> on the <em>set</em> underlying the 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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</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>, while the second function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∪</mo> <mi>X</mi> <mi>E</mi></msubsup></mrow><annotation encoding="application/x-tex">\cup^E_X</annotation></semantics></math> is the operation of forming the <a href="cup+product#CupProductInWhiteheadGeneralizedCohomologyTheory">E-cup product</a> of <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> of its elements.</p> <p>Both of these operations are clearly <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> (for the first this is evident, for the second this comes down to the fact that the <a href="suspension+spectrum#SmashMonoidalDiagonals">smash-monoidal diagonals of suspension spectra</a> are natural, hence are indeed <a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal diagonals</a>). Therefore the <a class="existingWikiWord" href="/nlab/show/full+subcategory">fully-faithfulness</a> of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>PointedTopologicalSpaces</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>A</mi><mo>↦</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mspace width="thickmathspace"></mspace><mi>Functors</mi><mo maxsize="1.8em" minsize="1.8em">(</mo><mi>Ho</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>PointedTopologicalSpaces</mi><msup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo maxsize="1.8em" minsize="1.8em">)</mo></mrow><annotation encoding="application/x-tex"> Ho \big( PointedTopologicalSpaces \big) \;\overset{ \;\; A \mapsto [-,A]\;\; }{\hookrightarrow}\; Functors \Big( Ho \big( PointedTopologicalSpaces \big)^{op} , Set \Big) </annotation></semantics></math></div> <p>implies that this natural transformation <a class="maruku-eqref" href="#eq:CupSquareInMultiplicativeCohomologyNaturalTransformation">(2)</a> between the <a class="existingWikiWord" href="/nlab/show/representable+functors">representable functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mi>n</mi></msup><mi>E</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,\Omega^\infty \Sigma^n E]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>Ω</mi> <mn>∞</mn></msup><msup><mi>Σ</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mi>E</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,\Omega^\infty \Sigma^{2n} E]</annotation></semantics></math> is itself represented by a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <a class="maruku-eqref" href="#eq:CupSquareAsUnstableCohomologyOperation">(1)</a> between the representing classifying spaces, hence by an unstable cohomology operation.</p> <p></p> </div> </p> <p>The analogous construction of course exists for 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>th cup power <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><msub><mi>k</mi> <mo>∪</mo></msub></mrow></msup></mrow><annotation encoding="application/x-tex">(-)^{k_\cup}</annotation></semantics></math> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math>. Brief mentioning of this is in <a href="#Wirthmuller12">Wirthmüller 12, Example (1) on p. 44 (46 of 67)</a>.</p> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>Steenrod’s original colloquium lectures:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>, <em>Cohomology operations, and obstructions to extending continuous functions</em>, Advances in Math. 8, 371–416. (1972) (<a href="https://doi.org/10.1016/0001-8708(72)90004-7">doi:10.1016/0001-8708(72)90004-7</a>, <a href="http://www.maths.ed.ac.uk/%7Eaar/papers/steen6.pdf">pdf</a>)</li> </ul> <p>Background and review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Michael+Boardman">John Michael Boardman</a>, <em>Stable Operations in Generalized Cohomology</em> [<a href="https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Boardman-StableOperations.pdf" title="pdf">pdf</a>] in: <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a> (ed.) <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em> Oxford 1995 (<a href="https://doi.org/10.1016/B978-0-444-81779-2.X5000-7">doi:10.1016/B978-0-444-81779-2.X5000-7</a>)</p> </li> <li id="May"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, chapter 22, section 5 of: <em>A concise course in algebraic topology</em>, University of Chicago Press 1999 (ISBN:978-0226511832, <a href="https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf">pdf</a>)</p> </li> </ul> <p>On the structure on the collection of all unstable cohomology operations:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Michael+Boardman">John Michael Boardman</a>, <a class="existingWikiWord" href="/nlab/show/David+Copeland+Johnson">David Copeland Johnson</a>, <a class="existingWikiWord" href="/nlab/show/W.+Stephen+Wilson">W. Stephen Wilson</a>, <em>Unstable Operations in Generalized Cohomology</em> (<a href="https://hopf.math.purdue.edu/Boardman-Johnson-Wilson/bjw.pdf">pdf</a>), in: <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a> (ed.) <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em> Oxford 1995 (<a href="https://doi.org/10.1016/B978-0-444-81779-2.X5000-7">doi:10.1016/B978-0-444-81779-2.X5000-7</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Stacey">Andrew Stacey</a>, <a class="existingWikiWord" href="/nlab/show/Sarah+Whitehouse">Sarah Whitehouse</a>, <em>The Hunting of the Hopf Ring</em>, Homology Homotopy Appl. Volume 11, Number 2 (2009), 75-132. (<a href="https://arxiv.org/abs/0711.3722">arXiv:0711.3722</a>, <a href="https://projecteuclid.org/euclid.hha/1251832594">euclid:hha/1251832594</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tilman+Bauer">Tilman Bauer</a>, <em>Formal plethories</em>, Advances in Mathematics Volume 254, 20 March 2014, Pages 497-569 (<a href="https://arxiv.org/abs/1107.5745">arXiv:1107.5745</a>, <a href="https://doi.org/10.1016/j.aim.2013.12.023">doi:10.1016/j.aim.2013.12.023</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/William+Mycroft">William Mycroft</a>, <em>Unstable Cohomology Operations: Computational Aspects of Plethories</em>, 2017 (<a href="https://etheses.whiterose.ac.uk/20524/1/wm-thesis-final.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MycroftUnstableCohomologyOperations.pdf" title="pdf">pdf</a>)</p> </li> </ul> <h3 id="on_ordinary_cohomology">On ordinary cohomology</h3> <p>Operations on <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+Mosher">Robert Mosher</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Tangora">Martin Tangora</a>, <em>Cohomology Operations and Application in Homotopy Theory</em>, Harper and Row (1968), Dover (2008) (<a href="https://store.doverpublications.com/0486466647.html">ISBN10:0486466647</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Stacey">Andrew Stacey</a>, <a class="existingWikiWord" href="/nlab/show/Sarah+Whitehouse">Sarah Whitehouse</a>, <em>Stable and unstable operations in mod <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> cohomology theories</em>, Algebr. Geom. Topol. Volume 8, Number 2 (2008), 1059-1091 (<a href="https://arxiv.org/abs/math/0605471">arXiv:math/0605471</a>, <a href="https://projecteuclid.org/euclid.agt/1513796856">euclid:agt/1513796856</a>)</p> </li> </ul> <h3 id="on_topological_ktheory">On topological K-theory</h3> <p>Operations on <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>:</p> <ul> <li id="Wirthmuller12"><a class="existingWikiWord" href="/nlab/show/Klaus+Wirthm%C3%BCller">Klaus Wirthmüller</a>, Sections 11 and 13 of: <em>Vector bundles and K-theory</em>, 2012 (<a class="existingWikiWord" href="/nlab/files/wirthmueller-vector-bundles-and-k-theory.pdf" title="pdf">pdf</a>)</li> </ul> <p>For <a class="existingWikiWord" href="/nlab/show/Adams+operations">Adams operations</a>:</p> <ul> <li id="AguilarGitlerPrieto">Marcelo Aguilar, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, Carlos Prieto, Section 10 of: <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href="https://link.springer.com/book/10.1007/b97586">doi:10.1007/b97586</a>)</li> </ul> <p>More:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/William+Mycroft">William Mycroft</a>, <a class="existingWikiWord" href="/nlab/show/Sarah+Whitehouse">Sarah Whitehouse</a>, <em>The plethory of operations in complex topological K-theory</em> (<a href="https://arxiv.org/abs/2001.01608">arXiv:2001.01608</a>)</li> </ul> <h3 id="on_complexoriented_cohomology_theories">On complex-oriented cohomology theories</h3> <p>For <a class="existingWikiWord" href="/nlab/show/Steenrod+operations">Steenrod operations</a> on <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology">complex-oriented</a> <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology">generalized cohomology</a> (such as <a class="existingWikiWord" href="/nlab/show/MU">MU</a> and <a class="existingWikiWord" href="/nlab/show/BP">BP</a>):</p> <ul> <li id="Kochman96"><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</a>, section 2.5 and 3.5 of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</li> </ul> <h3 id="on_differential_cohomology">On differential cohomology</h3> <p>Refinement to <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Primary operations in differential cohomology</em>, Adv. Math. 335 (2018), 519-562 (<a href="https://arxiv.org/abs/1604.05988">arXiv:1604.05988</a>, <a href="https://www.sciencedirect.com/science/article/pii/S0001870818302676?via%3Dihub">doi:10.1016/j.aim.2018.07.019</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 5, 2023 at 19:31:55. 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