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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="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <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> </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='#definition'>Definition</a></li> <li><a href='#Examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>Bockstein homomorphism</em> is a <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a> induced from a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> whose injective map is given by multiplication with an <a class="existingWikiWord" href="/nlab/show/integer">integer</a>.</p> <p>The archetypical examples are the Bockstein homomorphisms induced this way from the short exact sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mover><mo>→</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><mi>ℤ</mi><mover><mo>→</mo><mrow></mrow></mover><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \stackrel{}{\to} \mathbb{Z}/2\mathbb{Z} \,. </annotation></semantics></math></div> <p>These relate notably 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> <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> 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><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math> (such as <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+classes">Stiefel-Whitney classes</a>) to cohomology with integral coefficients in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> (such as <a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+classes">integral Stiefel-Whitney classes</a>).</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>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/integer">integer</a>. Then multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mrow><mi>m</mi><mo>⋅</mo></mrow></mover><mi>A</mi></mrow><annotation encoding="application/x-tex"> A \stackrel{m\cdot}{\to} A </annotation></semantics></math></div> <p>induces a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of abelian groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>A</mi><mo stretchy="false">/</mo><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>tors</mi></mrow></msub><mover><mo>→</mo><mrow><mi>m</mi><mo>⋅</mo></mrow></mover><mi>A</mi><mo>→</mo><mi>A</mi><mo stretchy="false">/</mo><mi>m</mi><mi>A</mi><mo>→</mo><mn>0</mn><mo>,</mo></mrow><annotation encoding="application/x-tex"> 0\to A/A_{m-tors} \stackrel{m\cdot}{\to} A \to A/m A\to 0, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>tors</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A_{m-tors}</annotation></semantics></math> is the subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/torsion+subgroup">torsion elements</a> of <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>, and so a long <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>tors</mi></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>m</mi><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>tors</mi></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \cdots \mathbf{B}^n (A/A_{m-tors}) \to \mathbf{B}^n A \to \mathbf B^n(A/ m A) \to \mathbf{B}^{n+1} (A/A_{m-tors}) \to \cdots </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>s, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n(-)</annotation></semantics></math> denotes the <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>-fold <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> (hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^n A</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+object">Eilenberg-MacLane object</a> 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> 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> <p>This induces, in turn, for any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> the ambient <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, such as <a class="existingWikiWord" href="/nlab/show/Top">Top</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>) , a long <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>tors</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>m</mi><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>β</mi> <mi>m</mi></msub></mrow></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>tors</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \cdots \mathbf{H}(X,\mathbf{B}^n (A/A_{m-tors})) \to \mathbf{H}(X,\mathbf{B}^n A) \to \mathbf{H}(X,\mathbf B^n(A/ m A)) \stackrel{\beta_m}{\to} \mathbf{H}(X,\mathbf{B}^{n+1} (A/A_{m-tors})) \to \cdots </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>.</p> <p>Here the <a class="existingWikiWord" href="/nlab/show/connecting+homomorphisms">connecting homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\beta_m</annotation></semantics></math> are called the <strong>Bockstein homomorphisms</strong>.</p> <p>Notice that often this term is used to refer only to the image of the above in <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, hence to the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">\beta_m</annotation></semantics></math> under <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncation</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+group">0th homotopy group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>m</mi></msub><mo>:</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">/</mo><mi>m</mi><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><msub><mi>A</mi> <mrow><mi>m</mi><mo>−</mo><mi>tors</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \beta_m : H^n(X,A/ m A) \to H^{n+1}(X,(A/A_{m-tors})) \,. </annotation></semantics></math></div> <h2 id="Examples">Examples</h2> <div class="num_example" id="Mod2BocksteinIntoIntegralCohomology"> <h6 id="example">Example</h6> <p><strong>(mod 2 Bockstein homomorphism into <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a>)</strong></p> <p>The Bockstein homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> for the sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mover><mo>⟶</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><mi>ℤ</mi><mover><mo>⟶</mo><mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></mover><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \mathbb{Z} \stackrel{\cdot 2}{\longrightarrow} \mathbb{Z} \stackrel{mod\, 2}{\longrightarrow} \mathbb{Z}/2\mathbb{Z} </annotation></semantics></math></div> <p>serves to define <a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+classes">integral Stiefel-Whitney classes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≔</mo><mi>β</mi><msub><mi>w</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> W_{n+1} \coloneqq \beta w_n </annotation></semantics></math></div> <p>in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math> from <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><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math>-valued <a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+classes">Stiefel-Whitney classes</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>.</p> </div> <div class="num_example" id="Mod2BocksteinIntoMod2Cohomology"> <h6 id="example_2">Example</h6> <p><strong>(first <a class="existingWikiWord" href="/nlab/show/Steenrod+square">Steenrod square</a>)</strong></p> <p>The Bockstein homomorphism for the sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mover><mo>⟶</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover><mi>ℤ</mi><mo stretchy="false">/</mo><mn>4</mn><mi>ℤ</mi><mover><mo>⟶</mo><mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></mover><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \mathbb{Z}/2\mathbb{Z} \overset{\cdot 2}{\longrightarrow} \mathbb{Z}/4\mathbb{Z} \overset{mod\, 2}{\longrightarrow} \mathbb{Z}/2\mathbb{Z} </annotation></semantics></math></div> <p>is also called the <em>first <a class="existingWikiWord" href="/nlab/show/Steenrod+square">Steenrod square</a></em>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">Sq^1</annotation></semantics></math>.</p> <p>This is often equivalently denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math>, as in example <a class="maruku-ref" href="#Mod2BocksteinIntoIntegralCohomology"></a>. The difference between the two is just the mod-2 reduction in their codomain:</p> <div class="maruku-equation" id="eq:Mod2BocksteinSequences"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover></mtd> <mtd><mi>ℤ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></mover></mtd> <mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mtd> <mtd><mover><mo>⟶</mo><mi>β</mi></mover></mtd> <mtd><mi>B</mi><mi>ℤ</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>4</mn></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></munder></mtd> <mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>4</mn><mi>ℤ</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></munder></mtd> <mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>4</mn><mi>ℤ</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msup><mi>Sq</mi> <mn>1</mn></msup></mrow></munder></mtd> <mtd><mi>B</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z} &\overset{\cdot 2}{\longrightarrow}& \mathbb{Z} &\overset{mod\, 2}{\longrightarrow}& \mathbb{Z}/2\mathbb{Z} &\simeq& \mathbb{Z}/2\mathbb{Z} &\overset{\beta}{\longrightarrow}& B \mathbb{Z} \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{mod\, 4}} && \downarrow^{\mathrlap{id}} && \downarrow^{ id } && \downarrow^{\mathrlap{B(mod\, 2)}} \\ \mathbb{Z}/2\mathbb{Z} &\underset{\cdot 2 }{\longrightarrow}& \mathbb{Z}/4\mathbb{Z} &\underset{mod\, 2}{\longrightarrow}& \mathbb{Z}/2\mathbb{Z} &\simeq& (\mathbb{Z}/4\mathbb{Z})/(\mathbb{Z}/2\mathbb{Z}) &\underset{Sq^1}{\longrightarrow}& B (\mathbb{Z}/2\mathbb{Z}) } </annotation></semantics></math></div> <p>More generally, for <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> any <a class="existingWikiWord" href="/nlab/show/prime+number">prime number</a> the multiplication by <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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><msup><mi>p</mi> <mn>2</mn></msup></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{p^2}</annotation></semantics></math> induces the short exact sequence <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><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mrow><msup><mi>p</mi> <mn>2</mn></msup></mrow><mi>ℤ</mi><mo>→</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mi>p</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/p\mathbb{Z} \to \mathbb{Z}/{p^2}\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z}</annotation></semantics></math>. The corresponding Bockstein homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\beta_p</annotation></semantics></math> appears as one of the generators of the 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> <a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a>.</p> </div> <div class="num_example" id="IntegralSteenrodSquares"> <h6 id="example_3">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/integral+Steenrod+squares">integral Steenrod squares</a>)</strong></p> <p>For <a class="existingWikiWord" href="/nlab/show/odd+natural+numbers">odd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">2n + 1 \in \mathbb{N}</annotation></semantics></math> defines the <a class="existingWikiWord" href="/nlab/show/integral+Steenrod+squares">integral Steenrod squares</a> to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Sq</mi> <mi>ℤ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>β</mi><mo>∘</mo><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sq^{2n + 1}_{\mathbb{Z}} \;\coloneqq\; \beta \circ Sq^{2n} \,. </annotation></semantics></math></div> <p>By example <a class="maruku-ref" href="#Mod2BocksteinIntoMod2Cohomology"></a> and by the first <a class="existingWikiWord" href="/nlab/show/Adem+relation">Adem relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mn>1</mn></msup><mo>∘</mo><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>=</mo><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">Sq^1 \circ Sq^{2n} = Sq^{2n+1}</annotation></semantics></math> (<a href="Steenrod+square#CompositionWithSq1">this example</a>) these indeed are lifts of the odd <a class="existingWikiWord" href="/nlab/show/Steenrod+squares">Steenrod squares</a>:</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><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo stretchy="false">)</mo><mo>∘</mo><msubsup><mi>Sq</mi> <mi>ℤ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (mod\, 2) \circ Sq^{2n + 1}_{\mathbb{Z}} \;=\; Sq^{2n+1} \,, </annotation></semantics></math></div> <p>because, by <a class="maruku-eqref" href="#eq:Mod2BocksteinSequences">(1)</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>Sq</mi> <mi>ℤ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>Sk</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow></mover></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>β</mi></mover></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>B</mi> <mrow><mi>k</mi><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msup><mi>Sk</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow></munder></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msup><mi>Sq</mi> <mn>1</mn></msup></mrow></munder></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Sq^{2n+1}_{\mathbb{Z}} &\colon& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{Sk^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{ \beta }{\longrightarrow}& B^{\bullet + 2n + 1} \mathbb{Z} \\ && \downarrow^{ id } && \downarrow^{ id } && \downarrow^{\mathrlap{B^{k + 2 n + 1}(mod\, 2)}} \\ Sq^{2n+1} &\colon& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{Sk^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{ Sq^1 }{\longrightarrow}& B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z}) } </annotation></semantics></math></div></div> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">A=\mathbb{Z}</annotation></semantics></math>, the equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">|</mo><mo>≅</mo><mo stretchy="false">|</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">\vert \mathbf{B}^{n+1}\mathbb{Z} \vert \cong \vert \mathbf{B}^n U(1)\vert</annotation></semantics></math> (which holds in ambient contexts such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/ETop%E2%88%9EGrpd">ETop∞Grpd</a> or <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> under <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>:</mo><mi>ETop</mi><mn>∞</mn><mi>Grpd</mi><mover><mo>→</mo><mi>Π</mi></mover><mn>∞</mn><mi>Grpd</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>Top</mi></mrow><annotation encoding="application/x-tex">\vert - \vert : ETop \infty Grpd \stackrel{\Pi}{\to} \infty Grpd \stackrel{\simeq}{\to} Top</annotation></semantics></math>) identifies the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mi>m</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}^n(\mathbb{Z}_m)\to \mathbf{B}^{n+1}\mathbb{Z}</annotation></semantics></math> with the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mi>m</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^n(\mathbb{Z}_m)\to \mathbf{B}^{n} U(1)</annotation></semantics></math> induced by the inclusion of the subgroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math>-th roots of unity into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>. This identifies the Bockstein homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mi>m</mi></msub><mo>:</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><msub><mi>ℤ</mi> <mi>m</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\beta_m: H^n(X;\mathbb{Z}_m)\to H^{n+1}(X;\mathbb{Z})</annotation></semantics></math> with the natural homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><msub><mi>ℤ</mi> <mi>m</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^n(X;\mathbb{Z}_m)\to H^{n}(X;U(1))</annotation></semantics></math>.</p> <p>More in detail:</p> <div class="num_example" id="Mod2BocksteinAndExponentialExactSequence"> <h6 id="example_4">Example</h6> <p><strong>(mod 2 <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a> and the <a class="existingWikiWord" href="/nlab/show/exponential+exact+sequence">exponential exact sequence</a>)</strong></p> <p>Let</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo>⟶</mo><mi>B</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\beta \;\colon\; \mathbb{Z}/2\mathbb{Z} \longrightarrow B \mathbb{Z}</annotation></semantics></math> be the ordinare Bockstein homomorphism</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo>≔</mo><mo stretchy="false">(</mo><mo>⋅</mo><mi>π</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo>↪</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota\coloneqq (\cdot \pi) \;\colon\; \mathbb{Z}/2\mathbb{Z} \hookrightarrow U(1)</annotation></semantics></math> the canonical inclusion;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⟶</mo><mi>B</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\delta \;\colon\; U(1) \longrightarrow B\mathbb{Z}</annotation></semantics></math> the classifying map.</p> </li> </ol> <p>Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>β</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>δ</mi><mo>∘</mo><mi>ι</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \beta \;=\; \delta \circ \iota \,. </annotation></semantics></math></div> <p>Because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>⋅</mo><mn>2</mn></mrow></mover></mtd> <mtd><mi>ℤ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></mover></mtd> <mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mtd> <mtd><mover><mo>⟶</mo><mi>β</mi></mover></mtd> <mtd><mi>B</mi><mi>ℤ</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>⋅</mo><mi>π</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>⋅</mo><mi>π</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><mi>ι</mi><mo>≔</mo><mpadded width="0"><mrow><mo>⋅</mo><mi>π</mi></mrow></mpadded></mrow></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>⋅</mo><mn>2</mn><mi>π</mi></mrow></munder></mtd> <mtd><mi>ℝ</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mi>π</mi></mrow></munder></mtd> <mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>ℝ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>π</mi><mi>ℤ</mi></mtd> <mtd><munder><mo>⟶</mo><mi>δ</mi></munder></mtd> <mtd><mi>B</mi><mi>ℤ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z} &\overset{\cdot 2}{\longrightarrow}& \mathbb{Z} &\overset{mod\, 2}{\longrightarrow}& \mathbb{Z}/2\mathbb{Z} &\simeq& \mathbb{Z}/2\mathbb{Z} &\overset{\beta}{\longrightarrow}& B \mathbb{Z} \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{\cdot \pi}} && \downarrow^{\mathrlap{\cdot \pi}} && \downarrow^{\iota \coloneqq \mathrlap{\cdot \pi}} && \downarrow^{\mathrlap{id}} \\ \mathbb{Z} &\underset{\cdot 2 \pi}{\longrightarrow}& \mathbb{R} &\underset{mod\, 2\pi}{\longrightarrow}& U(1) &\simeq& \mathbb{R}/2\pi\mathbb{Z} &\underset{\delta}{\longrightarrow}& B \mathbb{Z} } </annotation></semantics></math></div></div> <div class="num_prop" id="SteenrodSquaresAndDBCupProductOnOddClasses"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Deligne-Beilinson+cup+product">Deligne-Beilinson cup product</a> on odd-degree <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>)</strong></p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>H</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \hat H \;\colon\; X \longrightarrow \mathbf{B}^{2n} U(1)_{conn}) </annotation></semantics></math></div> <p>be a class in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a> with underlying class in odd degree</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><mi>H</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mover><mi>H</mi><mo stretchy="false">^</mo></mover></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mover><mo>⟶</mo><mi>χ</mi></mover><msup><mi>B</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> [H] \;\colon\; X \overset{\hat H}{\longrightarrow} \mathbf{B}^{n} U(1)_{conn} \overset{\chi}{\longrightarrow} B^{2n+1} \mathbb{Z} </annotation></semantics></math></div> <p>This implies that its <a class="existingWikiWord" href="/nlab/show/Beilinson-Deligne+cup+product">Beilinson-Deligne cup product</a> with itself satisfies</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>H</mi><mo stretchy="false">^</mo></mover><mover><mo>∪</mo><mo stretchy="false">^</mo></mover><mover><mi>H</mi><mo stretchy="false">^</mo></mover><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mover><mi>H</mi><mo stretchy="false">^</mo></mover><mover><mo>∪</mo><mo stretchy="false">^</mo></mover><mover><mi>H</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex"> \hat H \hat \cup \hat H = - \hat H \hat \cup \hat H </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>2</mn><mover><mi>H</mi><mo stretchy="false">^</mo></mover><mover><mo>∪</mo><mo stretchy="false">^</mo></mover><mover><mi>H</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>0</mn></mrow><annotation encoding="application/x-tex"> 2 \hat H \hat \cup \hat H \;\simeq\; 0 </annotation></semantics></math></div> <p>hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo><mo>∪</mo><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mn>0</mn></mrow><annotation encoding="application/x-tex"> 2 [H] \cup [H] \;\simeq\; 0 </annotation></semantics></math></div> <p>hence that the ordinary <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo><mo>∪</mo><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[H] \cup [H]</annotation></semantics></math> is a 2-torsion class. Let then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>j</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mover><mo>↪</mo><mrow><msup><mi>B</mi> <mrow><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>ι</mi><mo stretchy="false">)</mo></mrow></mover><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⟶</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> j \;\colon\; \mathbf{B}^{4n+1} \mathbb{Z}/2\mathbb{Z} \overset{ B^{4n+1} (\iota) }{\hookrightarrow} \flat \mathbf{B}^{4n+1} U(1) \longrightarrow \mathbf{B}^{4n+1} U(1)_{conn} </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi></mrow><annotation encoding="application/x-tex">\iota</annotation></semantics></math> from example <a class="maruku-ref" href="#Mod2BocksteinAndExponentialExactSequence"></a>.</p> <p>Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>H</mi><mo stretchy="false">^</mo></mover><mover><mo>∪</mo><mo stretchy="false">^</mo></mover><mover><mi>H</mi><mo stretchy="false">^</mo></mover><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>j</mi><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>H</mi><msub><mo stretchy="false">]</mo> <mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \hat H \hat \cup \hat H \;\simeq\; j Sq^{2n}([H]_{mod\,2}) \,. </annotation></semantics></math></div> <p>This is a <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>-refinement of the first <a class="existingWikiWord" href="/nlab/show/Adem+relation">Adem relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mn>1</mn></msup><mo>∘</mo><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>=</mo><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">Sq^1 \circ Sq^{2n} = Sq^{2n+1}</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/Steenrod+squares">Steenrod squares</a> (<a href="Steenrod+square#CompositionWithSq1">this example</a>) in that, by example <a class="maruku-ref" href="#Mod2BocksteinAndExponentialExactSequence"></a>, its image in ordinary cohomology with coefficients in <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><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2\mathbb{Z}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo><mo>∪</mo><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo><msub><mo stretchy="false">)</mo> <mrow><mi>mod</mi><mn>2</mn></mrow></msub></mtd> <mtd><mo>≃</mo></mtd> <mtd><munder><munder><mrow><msup><mi>Sq</mi> <mn>1</mn></msup></mrow><mo>⏟</mo></munder><mi>β</mi></munder><mo>∘</mo><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>H</mi><msub><mo stretchy="false">]</mo> <mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>=</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mi>H</mi><msub><mo stretchy="false">]</mo> <mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></msub><mo>∪</mo><mo stretchy="false">[</mo><mi>H</mi><msub><mo stretchy="false">]</mo> <mrow><mi>mod</mi><mo>,</mo><mn>2</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo>=</mo></mtd></mtr> <mtr><mtd><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>H</mi><msub><mo stretchy="false">]</mo> <mrow><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ ([H] \cup [H])_{mod 2} & \simeq & \underset{ \beta }{ \underbrace{ Sq^1 }} \circ Sq^{2n}([H]_{mod\,2}) \\ = \\ [H]_{mod\, 2} \cup [H]_{mod,2} \\ = \\ Sq^{2n+1}([H]_{mod\, 2}) } \,. </annotation></semantics></math></div></div> <p>This was first observed in (<a href="#Gomi08">Gomi 08</a>). Streamlined proofs are given in (<a href="#Bunke12">Bunke 12, propblem 3.106</a>, <a href="#GradySati16">Grady-Sati 16, prop. 22</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod+squares">Steenrod squares</a>, <a class="existingWikiWord" href="/nlab/show/cohomology+operation">cohomology operation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bockstein+spectral+sequence">Bockstein spectral sequence</a></p> </li> </ul> <h2 id="references">References</h2> <p>Original references include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Meyer+Bockstein">Meyer Bockstein</a>,</p> <p><em>Universal systems of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>-homology rings</em>, C. R. (Doklady) Acad. Sci. URSS (N.S.) 37 (1942), 243–245, MR0008701</p> <p><em>A complete system of fields of coefficients for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>-homological dimension</em> , C. R. (Doklady) Acad. Sci. URSS (N.S.) (1943), 38: 187–189, MR0009115</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Meyer+Bockstein">Meyer Bockstein</a>, <em>Sur la formule des coefficients universels pour les groupes d’homologie</em> , Comptes Rendus de l’académie des Sciences. Série I. Mathématique (1958), 247: 396–398, MR0103918</p> </li> </ul> <p>The relation to the <a class="existingWikiWord" href="/nlab/show/Beilinson-Deligne+cup+product">Beilinson-Deligne cup product</a> is discussed in</p> <ul> <li id="Gomi08"> <p><a class="existingWikiWord" href="/nlab/show/Kiyonori+Gomi">Kiyonori Gomi</a>, <em>Differential characters and the Steenrod squares</em>, In Groups of diffeomorphisms, volume 52 of Adv. Stud. Pure Math., pages 297?308. Math. Soc. Japan, Tokyo, 2008</p> </li> <li id="Bunke12"> <p><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, problem 3.106 in <em>Differential cohomology</em> (<a href="https://arxiv.org/abs/1208.3961">arXiv:1208.3961</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, prop. 22 in: <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>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 31, 2021 at 10:56:41. 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