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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> fiber integration </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/1139/#Item_13" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/measure">measure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/virtual+fundamental+class">virtual</a>) <a class="existingWikiWord" href="/nlab/show/fundamental+class">fundamental class</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemann+integration">Riemann</a>/<a class="existingWikiWord" href="/nlab/show/Lebesgue+integration">Lebesgue integration</a> <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">of differential forms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward in generalized cohomology</a>/<a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">in differential cohomology</a></td></tr> </tbody></table> <h2 id="analytic_integration">Analytic integration</h2> <p><a class="existingWikiWord" href="/nlab/show/integral+calculus">integral calculus</a></p> <p><a class="existingWikiWord" href="/nlab/show/Riemann+integration">Riemann integration</a>, <a class="existingWikiWord" href="/nlab/show/Lebesgue+integration">Lebesgue integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/line+integral">line integral</a>/<a class="existingWikiWord" href="/nlab/show/contour+integration">contour integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a>, <a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a>, <a class="existingWikiWord" href="/nlab/show/fermionic+path+integral">fermionic path integral</a></p> <p><a class="existingWikiWord" href="/nlab/show/Kontsevich+integral">Kontsevich integral</a>, <a class="existingWikiWord" href="/nlab/show/Selberg+integral">Selberg integral</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+Selberg+integral">elliptic Selberg integral</a></p> <h2 id="cohomological_integration">Cohomological integration</h2> <p><a class="existingWikiWord" href="/nlab/show/integration+in+ordinary+differential+cohomology">integration in ordinary differential cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/integration+in+differential+K-theory">integration in differential K-theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/Riemann-Roch+theorem">Riemann-Roch theorem</a></p> <h2 id="variants">Variants</h2> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></p> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a></p> <p><a class="existingWikiWord" href="/nlab/show/Batalin-Vilkovisky+integral">Batalin-Vilkovisky integral</a></p></div></div> <h4 id="cohomology">Cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>, <a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a>, <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </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> <ul> <li><a href='#fiberwise_integration_of_ordinary_differential_forms'>Fiberwise integration of ordinary differential forms</a></li> <li><a href='#in_generalized_cohomology_via_pontryaginthom_collapse_maps'>In generalized cohomology via Pontryagin-Thom collapse maps</a></li> <ul> <li><a href='#along_maps_of_manifolds'>Along maps of manifolds</a></li> <li><a href='#along_representable_morphisms_of_stacks'>Along representable morphisms of stacks</a></li> </ul> <li><a href='#InGeneralizedCohomologyByUmkehrMaps'>In generalized cohomology by Umkehr maps via abstract duality</a></li> <ul> <li><a href='#abstract_duality_and_atiyahmilnorspanier_duality__pontryaginthom_collapse'>Abstract duality and Atiyah-Milnor-Spanier duality + Pontryagin-Thom collapse</a></li> <li><a href='#in_linear_homotopytype_theory'>In linear homotopy-type theory</a></li> </ul> <li><a href='#in_generalized_differential_cohomology'>In generalized differential cohomology</a></li> <li><a href='#InKKTheory'>In KK-theory</a></li> <ul> <li><a href='#KKPushForwardAlongEmbedding'>Along an embedding</a></li> <li><a href='#KKPushforwardAlongSubmersion'>Along a proper submersion</a></li> <li><a href='#AlongAFibrationOfClosedSpinCManifolds'>Along a smooth fibration of closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-manifolds</a></li> <li><a href='##KKPushforwardAlongGeneralMap'>Along a general K-oriented map</a></li> <li><a href='#KKPushforwardAlongGeneralMap'>In twisted K-theory</a></li> </ul> <li><a href='#of_cohesive_differential_form_data'>Of cohesive differential form data</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#to_the_point'>To the point</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#ReferencesInKKTheory'>In noncommutative topology and KK-theory</a></li> <li><a href='#abstract_formulation'>Abstract formulation</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Fiber integration</em> or <em>push-forward</em> is a process that sends <a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a> classes on a <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">E \to B</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> to cohomology classes on the base <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> of the bundle, by <em>evaluating them on each fiber</em> in some sense.</p> <p>This sense is such that if the cohomology in question is <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> then fiber integration is ordinary <a class="existingWikiWord" href="/nlab/show/integration">integration</a> of <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>s over the fibers. Generally, the fiber integration over a bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-dimensional fibers reduces the degree of the cohomology class by <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>.</p> <p>Composing pullback of cohomology classes with fiber integration yields the notion of <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a>.</p> <h2 id="definition">Definition</h2> <h3 id="fiberwise_integration_of_ordinary_differential_forms">Fiberwise integration of ordinary differential forms</h3> <p>Consider a smooth <a class="existingWikiWord" href="/nlab/show/submersion">submersion</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon T\to B</annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>, whose fibers have dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>. The fiberwise integration of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> is a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo lspace="verythinmathspace">:</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mi>d</mi></mrow></msup><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_*\colon\Omega^n(T)\to\Omega^{n-d}(B)</annotation></semantics></math></div> <p>defined as follows. Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega\in\Omega^n(T)</annotation></semantics></math>, the differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-d)</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_*(\omega)</annotation></semantics></math> is constructed as follows.</p> <p>The value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_*(\omega)</annotation></semantics></math> at some collection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-d)</annotation></semantics></math> tangent vector fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v_1</annotation></semantics></math>, …, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mrow><mi>n</mi><mo>−</mo><mi>d</mi></mrow></msub></mrow><annotation encoding="application/x-tex">v_{n-d}</annotation></semantics></math> is computed as follows.</p> <p>First, lift each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i</annotation></semantics></math> to a section of the quotient bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(T)/T(f)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(f)</annotation></semantics></math> is the <span class="newWikiWord">relative tangent bundle<a href="/nlab/new/relative+tangent+bundle">?</a></span>. Lift these sections to sections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">u_i</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(T)</annotation></semantics></math> in some arbitrary way. Substitute the resulting vector fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> into the differential <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>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, obtaining a differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math>. Pull back <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> to each of the fibers of the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>T</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f\colon T\to B</annotation></semantics></math>, obtaining a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-form on each of these fibers, which does not depend on the choices of liftings <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">u_i</annotation></semantics></math>. Now integrate the resulting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>-form over each of these fibers. This gives a number for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">b\in B</annotation></semantics></math>, which depends smoothly on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>. This is the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_*(\omega)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v_1</annotation></semantics></math>, …, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mrow><mi>n</mi><mo>−</mo><mi>d</mi></mrow></msub></mrow><annotation encoding="application/x-tex">v_{n-d}</annotation></semantics></math>.</p> <p>See Greub, Halperin, and Vanstone, Volume I, Section VII.7.12.</p> <h3 id="in_generalized_cohomology_via_pontryaginthom_collapse_maps">In generalized cohomology via Pontryagin-Thom collapse maps</h3> <h4 id="along_maps_of_manifolds">Along maps of manifolds</h4> <p>Here is the rough outline of the construction via <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+collapse+maps">Pontryagin-Thom collapse maps</a>.</p> <p>The basic strategy is this:</p> <ol> <li> <p>start with a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">E \to B</annotation></semantics></math></p> </li> <li> <p>make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> bigger without changing its homotopy type such that the map from <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> becomes an <a class="existingWikiWord" href="/nlab/show/embedding">embedding</a>;</p> </li> <li> <p>choose an <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation</a> structure that makes the cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> equivalent to that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(E)</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a>);</p> </li> <li> <p>compose the Thom isomorphism with the pullback along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>Th</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \to Th(E)</annotation></semantics></math> to get an “Umkehr” map from cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> to cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> </li> </ol> <p>Now in detail.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p : E \to B</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> of smooth compact <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> with typical <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>.</p> <p>By the <a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a> one can choose an embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>:</mo><mi>E</mi><mo>↪</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">e:E \hookrightarrow \mathbb{R}^n</annotation></semantics></math> for some <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>. From this one obtains an embedding</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>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo><mo>:</mo><mi>E</mi><mo>↪</mo><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (p,e) : E \hookrightarrow B \times \mathbb{R}^n \,. </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_{(p,e)} (E)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> relative to this embedding. It is a rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mi>dim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex">n- dim F</annotation></semantics></math> bundle over the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">B \times \mathbb{R}^n</annotation></semantics></math>.</p> <p>Fix a <a class="existingWikiWord" href="/nlab/show/tubular+neighbourhood">tubular neighbourhood</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">B \times \mathbb{R}^n</annotation></semantics></math> and identify it with the total space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">N_{(p,e)}</annotation></semantics></math>. Then collapsing the whole <a class="existingWikiWord" href="/nlab/show/complement">complement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∖</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \times \mathbb{R}^n \setminus N_{(p,e)}(E)</annotation></semantics></math> to a point gives the <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_{(p,e)}(E)</annotation></semantics></math>, and the quotient map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>−</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Th</mi><mo stretchy="false">(</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> B \times \mathbb{R}^n \to B \times \mathbb{R}^n / (B \times \mathbb{R}^n - N_{(p,e)}(E)) \simeq Th(N_{(p,e)}(E)) </annotation></semantics></math></div> <p>factors through the <a class="existingWikiWord" href="/nlab/show/one-point+compactification">one-point compactification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">(B \times \mathbb{R}^n)^*</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">B \times \mathbb{R}^n</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>≅</mo><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>B</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">(B \times \mathbb{R}^n)^*\cong \Sigma^n B_+</annotation></semantics></math> (see <a href="one-point+compactification#MonoidalFunctoriality">here</a>), 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/suspension">suspension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">B_+</annotation></semantics></math> (or, equivalently, the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> with 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>-sphere: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>B</mi> <mo>+</mo></msub><mo>=</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>∧</mo><msub><mi>B</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\Sigma^n B_+= S^n \wedge B_+</annotation></semantics></math>), we obtain a factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>B</mi> <mo>+</mo></msub><mover><mo>→</mo><mi>τ</mi></mover><mi>Th</mi><mo stretchy="false">(</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> B \times \mathbb{R}^n \to \Sigma^n B_+ \stackrel{\tau}{\to} Th(N_{(p,e)}(E)) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> is called the <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+collapse+map">Pontrjagin-Thom collapse map</a>.</p> <p>Explicitly, as sets we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>B</mi> <mo>+</mo></msub><mo>≃</mo><mi>B</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Sigma^n B_+ \simeq B \times \mathbb{R}^n \cup \{\infty\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">Th(N_{(e,p)}(E)) = N_{(e,p)} \cup \{\infty\}</annotation></semantics></math>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>B</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">U \subset \Sigma^n B_+</annotation></semantics></math> a tubular neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>U</mi><mo>→</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi : U \to N_{(e,p)}(E)</annotation></semantics></math> an isomorphism, the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>:</mo><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>B</mi> <mo>+</mo></msub><mover><mo>→</mo><mrow></mrow></mover><mi>Th</mi><mo stretchy="false">(</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau : \Sigma^n B_+ \stackrel{}{\to} Th(N_{(p,e)}(E)) </annotation></semantics></math></div> <p>is defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>:</mo><mi>x</mi><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>x</mi><mo>∈</mo><mi>U</mi></mtd></mtr> <mtr><mtd><mn>∞</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau : x \mapsto \left\{ \array{ \phi(x) & | & x \in U \\ \infty & | & otherwise } \right. \,. </annotation></semantics></math></div> <p>Now let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> be some <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a>, and assume that the Thom space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Th</mi><mo stretchy="false">(</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Th(N_{(p,e)}(E))</annotation></semantics></math> has an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation</a>, so that we have a <a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a>. Then combined with the <a class="existingWikiWord" href="/nlab/show/suspension+isomorphism">suspension isomorphism</a> the pullback along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> produces a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>F</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>dim</mi><mi>F</mi></mrow></msup><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int_F \; \colon \; H^\bullet(E) \longrightarrow H^{\bullet - dim F}(B) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/cohomology+rings">cohomology rings</a>:</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><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mo>≃</mo> <mrow><mi>Thom</mi><mspace width="thickmathspace"></mspace><mi>iso</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mover><mi>H</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>n</mi><mo>−</mo><mi>dim</mi><mi>F</mi></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Th</mi><mo stretchy="false">(</mo><msub><mi>N</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msubsup><mi>τ</mi> <mrow><mi>Pontrjagin</mi><mo>−</mo><mi>Thom</mi><mspace width="thickmathspace"></mspace><mi>collapse</mi></mrow> <mo>*</mo></msubsup></mrow></mpadded></mtd></mtr> <mtr><mtd><msup><mover><mi>H</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>n</mi><mo>−</mo><mi>dim</mi><mi>F</mi></mrow></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>B</mi> <mo>+</mo></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mo>≃</mo> <mrow><mi>suspension</mi><mspace width="thickmathspace"></mspace><mi>iso</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>dim</mi><mi>F</mi></mrow></msup><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ H^\bullet(E) \\ \big\downarrow {}^{ \mathrlap{ \simeq_{Thom\;iso} } } \\ \tilde H^{ \bullet + n - dim F } \big( Th(N_{(p,e)}(E)) \big) \\ \big\downarrow \mathrlap{ \tau^\ast_{ Pontrjagin-Thom\;collapse } } \\ \tilde H^{\bullet + n - dim F} \big( \Sigma^n B_+ \big) \\ \big\downarrow {}^\mathrlap{ \simeq_{suspension\;iso} } \\ H^{\bullet - dim F}(B) } \,. </annotation></semantics></math></div> <p>This operation is independent of the choices involved. It is the <strong>fiber integration</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>-cohomology along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">p : E \to B</annotation></semantics></math>.</p> <h4 id="along_representable_morphisms_of_stacks">Along representable morphisms of stacks</h4> <p>The above definition generalizes to one of push-forward in generalized cohomology on <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a> over <a class="existingWikiWord" href="/nlab/show/SmthMfd">SmthMfd</a> along <a class="existingWikiWord" href="/nlab/show/representable+morphisms+of+stacks">representable morphisms of stacks</a>.</p> <p>(…)</p> <h3 id="InGeneralizedCohomologyByUmkehrMaps">In generalized cohomology by Umkehr maps via abstract duality</h3> <p>We discuss now a general abstract reformulation in terms of <a class="existingWikiWord" href="/nlab/show/duality">duality</a> in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> of the above traditional constructions.</p> <h4 id="abstract_duality_and_atiyahmilnorspanier_duality__pontryaginthom_collapse">Abstract duality and Atiyah-Milnor-Spanier duality + Pontryagin-Thom collapse</h4> <div class="num_defn" id="SpanierDualityOperation"> <h6 id="definition_2">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>≔</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>∨</mo></msup><mo>∘</mo><msubsup><mi>Σ</mi> <mo>+</mo> <mn>∞</mn></msubsup><mo>≔</mo><msub><mi>L</mi> <mi>whe</mi></msub><mi>Top</mi><mo>→</mo><mi>𝕊</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> D \coloneqq (-)^\vee\circ \Sigma^\infty_+ \coloneqq L_{whe} Top \to \mathbb{S}Mod </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a> map which sends a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> first to its <a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a> and then that to its <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+spectra">(∞,1)-category of spectra</a>.</p> </div> <p>(<a href="#ABG11">ABG 11, def 10.3</a>).</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+manifold">compact manifold</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X \to \mathbb{R}^n</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/embedding">embedding</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>X</mi> <mrow><msub><mi>ν</mi> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">S^n \to X^{\nu_n}</annotation></semantics></math> for the classical <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+collapse+map">Pontryagin-Thom collapse map</a> for this situation, and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mo>→</mo><msup><mi>X</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi><mi>X</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathbb{S} \to X^{-T X} </annotation></semantics></math></div> <p>for the corresponding <a class="existingWikiWord" href="/nlab/show/looping">looping</a> map from the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> to the <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> of the negative <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</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>. Then <a class="existingWikiWord" href="/nlab/show/Atiyah+duality">Atiyah duality</a> produces an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi><mi>X</mi></mrow></msup><mo>≃</mo><mi>D</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> X^{- T X} \simeq D X </annotation></semantics></math></div> <p>which identifies the <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> with the <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Σ</mi> <mo>+</mo> <mn>∞</mn></msubsup><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma^\infty_+ X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\mathbb{S} Mod</annotation></semantics></math> and this constitutes a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></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></mtd> <mtd></mtd> <mtd><msup><mi>X</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi><mi>X</mi></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝕊</mi></mtd> <mtd><munder><mo>→</mo><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>X</mi><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>D</mi><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && X^{- T X} \\ & \nearrow & \downarrow^{\mathrlap{\simeq}} \\ \mathbb{S} &\underset{D(X \to \ast)}{\to}& D X } </annotation></semantics></math></div> <p>identifying the classical <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+collapse+map">Pontryagin-Thom collapse map</a> with the abstract <a class="existingWikiWord" href="/nlab/show/dual+morphism">dual morphism</a> construction of prop. <a class="maruku-ref" href="#SpanierDualityOperation"></a>.</p> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">W \hookrightarrow X</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/embedding">embedding</a> of <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a>, then <a class="existingWikiWord" href="/nlab/show/Atiyah+duality">Atiyah duality</a> identifies the <a class="existingWikiWord" href="/nlab/show/Pontryagin-Thom+collapse+maps">Pontryagin-Thom collapse maps</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mo>→</mo><msup><mi>X</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi><mi>X</mi></mrow></msup><mo>→</mo><msup><mi>W</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>T</mi><mi>W</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathbb{S} \to X^{-T X} \to W^{- T W} </annotation></semantics></math></div> <p>with the abstract <a class="existingWikiWord" href="/nlab/show/dual+morphisms">dual morphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mo>→</mo><mi>D</mi><mi>X</mi><mo>→</mo><mi>D</mi><mi>W</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{S} \to D X \to D W \,. </annotation></semantics></math></div></div> <p>(<a href="#ABG11">ABG 11, prop. 10.5</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Given now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><msub><mi>CRing</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E \in CRing_\infty</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, then the <a class="existingWikiWord" href="/nlab/show/dual+morphism">dual morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi><mo>→</mo><mi>D</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{S} \to D X</annotation></semantics></math> induces under <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> a similar Pontryagin-Thom collapse map, but now not in <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-modules">(∞,1)-modules</a> but in <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/%28%E2%88%9E%2C1%29-modules">(∞,1)-modules</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>D</mi><mi>X</mi><msub><mo>⊗</mo> <mi>𝕊</mi></msub><mi>E</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E \to D X \otimes_{\mathbb{S}} E \,. </annotation></semantics></math></div> <p>The image of this under the <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/generalized+cohomology+theory">cohomology</a> functor produces</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>D</mi><mi>X</mi><msub><mo>⊗</mo> <mi>𝕊</mi></msub><mi>E</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo><mo>→</mo><mi>E</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [D X \otimes_{\mathbb{S}} E, E] \to E \,. </annotation></semantics></math></div> <p>If now one has a <a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a> (<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/orientation+in+generalized+cohomology">orientation</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mi>X</mi><msub><mo>⊗</mo> <mi>𝕊</mi></msub><mi>E</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [D X \otimes_{\mathbb{S}} E, E] \simeq [X,E]</annotation></semantics></math> that identifies the cohomology of the dual object with the original cohomology, then together with produces the <strong>Umkehr map</strong></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>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>D</mi><mi>X</mi><msub><mo>⊗</mo> <mi>𝕊</mi></msub><mi>E</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> [X,E] \simeq [D X \otimes_{\mathbb{S}} E, E] \to E </annotation></semantics></math></div> <p>that pushes the <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 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> to the <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 of the point. Analogously if instead of the terminal map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to \ast</annotation></semantics></math> we start with a more general map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math>.</p> <p>More generally a <a class="existingWikiWord" href="/nlab/show/Thom+isomorphism">Thom isomorphism</a> may not exists, but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>D</mi><mi>X</mi><msub><mo>⊗</mo> <mi>𝕊</mi></msub><mi>E</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[D X \otimes_{\mathbb{S}} E, E]</annotation></semantics></math> may still be equivalent to a <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>-variant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><msub><mo stretchy="false">]</mo> <mi>χ</mi></msub></mrow><annotation encoding="application/x-tex">[X,E]_{\chi}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,E]</annotation></semantics></math>, namely to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>χ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>E</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Gamma_X(\chi),E]</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo lspace="verythinmathspace">:</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>E</mi><mi>Line</mi><mo>↪</mo><mi>E</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\chi \colon \Pi(X) \to E Line \hookrightarrow E Mod</annotation></semantics></math> is an (<a class="existingWikiWord" href="/nlab/show/flat+%28%E2%88%9E%2C1%29-bundle">flat</a>) <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/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>≃</mo><munder><mi>lim</mi><mo>→</mo></munder></mrow><annotation encoding="application/x-tex">\Gamma \simeq \underset{\to}{\lim}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> (the <a class="existingWikiWord" href="/nlab/show/generalized+Thom+spectrum">generalized Thom spectrum</a> construction). In this case the above yields a <em><a class="existingWikiWord" href="/nlab/show/twisted+Umkehr+map">twisted Umkehr map</a></em>.</p> </div> <p>(<a href="#ABG10">ABG 10, 9.1</a>)</p> <h4 id="in_linear_homotopytype_theory">In linear homotopy-type theory</h4> <p>We may formulate the above still a bit more abstractly in <a class="existingWikiWord" href="/nlab/show/linear+homotopy-type+theory">linear homotopy-type theory</a> (following <em><a class="existingWikiWord" href="/schreiber/show/Homotopy-type+semantics+for+quantization">Homotopy-type semantics for quantization</a></em>: see at <a class="existingWikiWord" href="/nlab/show/indexed+monoidal+infinity-category">indexed monoidal infinity-category</a> the section on <em><a href="indexed+monoidal+infinity-category#FundamentalClasses">Fundamental classes</a></em> and following.</p> <p><br /></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> is conjecturally <a class="existingWikiWord" href="/nlab/show/categorical+semantics">∞-categorical</a> <a class="existingWikiWord" href="/nlab/show/semantics">semantics</a> of <a class="existingWikiWord" href="/nlab/show/linear+homotopy+type+theory">linear homotopy type theory</a></strong>:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/linear+homotopy+type+theory">linear homotopy type theory</a></th><th><a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/quantum+theory">quantum theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+type">linear type</a></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/module+spectrum">module</a>-)<a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/multiplicative+conjunction">multiplicative conjunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/composite+system">composite system</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+linear+type">dependent linear type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-module+bundle">module spectrum bundle</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/six+operation+yoga">six operation yoga</a> in <a class="existingWikiWord" href="/nlab/show/Wirthm%C3%BCller+context">Wirthmüller context</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dual+object">dual type</a> (linear negation)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/invertible+type">invertible type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+generalized+cohomology">twist</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> (“<a class="existingWikiWord" href="/nlab/show/bra-ket">bra</a>”)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dual+type">dual</a> of <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> (“<a class="existingWikiWord" href="/nlab/show/bra-ket">ket</a>”)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bivariant+cohomology">bivariant cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantum+operators">quantum operators</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/exponential+modality">exponential modality</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> over <a class="existingWikiWord" href="/nlab/show/finite+homotopy+type">finite homotopy type</a> (of twist)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a> (<a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a>)</td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable</a> <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> over <a class="existingWikiWord" href="/nlab/show/finite+homotopy+type">finite homotopy type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Atiyah+duality">Atiyah duality</a> between <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> and <a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">(twisted) <a class="existingWikiWord" href="/nlab/show/self-dual+object">self-dual type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/inner+product">inner product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> coinciding with <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ambidextrous+adjunction">ambidexterity</a>, <a class="existingWikiWord" href="/nlab/show/semiadditive+%28%E2%88%9E%2C1%29-category">semiadditivity</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> coinciding with <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> up to <a class="existingWikiWord" href="/nlab/show/invertible+type">invertible type</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Wirthm%C3%BCller+isomorphism">Wirthmüller isomorphism</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>f</mi></msub><mo>⊣</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\sum_f \dashv f^\ast)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a></td><td style="text-align: left;">pushforward in <a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">(twisted-)<a class="existingWikiWord" href="/nlab/show/self-dual+object">self-duality</a>-<a href="self-dual+object#RelationToDaggerCompactStructure">induced dagger</a> of this counit</td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/twisted+Umkehr+map">twisted</a>-)<a class="existingWikiWord" href="/nlab/show/Umkehr+map">Umkehr map</a>/<a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+polynomial+functor">linear polynomial functor</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/correspondence">correspondence</a></td><td style="text-align: left;">space of <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+polynomial+functor">linear polynomial functor</a> with <a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+kernel">integral kernel</a> (<a class="existingWikiWord" href="/nlab/show/pure+motive">pure motive</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/prequantized+Lagrangian+correspondence">prequantized Lagrangian correspondence</a>/<a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></td></tr> <tr><td style="text-align: left;">composite of this <a class="existingWikiWord" href="/nlab/show/linear+implication">linear implication</a> with daggered-counit followed by unit</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+transform">integral transform</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/motivic+quantization">motivic</a>/cohomological <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trace">trace</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euler+characteristic">Euler characteristic</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a></td></tr> </tbody></table> </div> <p>(…)</p> <h3 id="in_generalized_differential_cohomology">In generalized differential cohomology</h3> <p>See</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a></p> </li> </ul> </li> </ul> <h3 id="InKKTheory">In KK-theory</h3> <p>We discuss fiber integration/push-forward/<a class="existingWikiWord" href="/nlab/show/Gysin+maps">Gysin maps</a> in <a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a>, hence in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> (<a href="#ConnesSkandalis84">Connes-Skandalis 85</a>, <a href="#BMRS07">BMRS 07, section 3</a>). For more see at <em><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+K-theory">fiber integration in K-theory</a>.</em></p> <p>The following discusses KK-pushforward</p> <ol> <li> <p><em><a href="#KKPushForwardAlongEmbedding">Along an embedding</a></em></p> </li> <li> <p><em><a href="#KKPushforwardAlongSubmersion">Along a submersion</a></em></p> </li> <li> <p><em><a href="#AlongAFibrationOfClosedSpinCManifolds">Along a fibration of closed spin^c manifolds</a></em></p> </li> <li> <p><em><a href="#KKPushforwardAlongGeneralMap">Along a general K-oriented map</a></em></p> </li> <li> <p><em><a href="#KKPushforwardAlongGeneralMap">In twisted K-theory</a></em></p> </li> </ol> <p>The construction goes back to (<a href="#Connes82">Connes 82</a>), where it is given over smooth manifolds. Then (<a href="#ConnesSkandalis84">Connes-Skandalis 84</a>, <a href="#HilsumSkandalis87">Hilsum-Skandalis 87</a>) generalize this to maps between <a class="existingWikiWord" href="/nlab/show/foliations">foliations</a> by KK-elements betwen the <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebras">groupoid convolution algebras</a> of the coresponding <a class="existingWikiWord" href="/nlab/show/holonomy+groupoids">holonomy groupoids</a> and (<a href="#RouseWang10">Rouse-Wang 10</a>) further generalize to the case where a <a class="existingWikiWord" href="/nlab/show/circle+2-bundle">circle 2-bundle</a> twist is present over these foliations. A purely algebraic generalization to (K-oriented) maps between otherwise arbitrary <a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative spaces</a>/<a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> is in (<a href="#BMRS07">BMRS 07</a>).</p> <h4 id="KKPushForwardAlongEmbedding">Along an embedding</h4> <p>(<a href="ConnesSkandalis84">Connes-Skandalis 84, above prop. 2.8</a>)</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">h \colon X \hookrightarrow Y</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/embedding">embedding</a> of <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>.</p> <p>The push-forward constructed from this is supposed to be an element in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>!</mo><mo lspace="verythinmathspace">:</mo><msub><mi>KK</mi> <mi>d</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> h! \colon KK_d(C(X), C(Y)) </annotation></semantics></math></div> <p>in terms of which the push-forward on <a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a> is induced by postcomposition:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mo>!</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>K</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>KK</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>h</mi><mo>!</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><msub><mi>KK</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>d</mi></mrow></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>KK</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>d</mi></mrow></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> h_! \;\colon\; K^\bullet(X) \simeq KK_\bullet(\mathbb{C}, X) \stackrel{h!\circ (-)}{\to} KK_{\bullet+d}(\mathbb{C},Y) \simeq KK^{\bullet+d}(Y) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>−</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d = dim(X) - dim(Y)</annotation></semantics></math>.</p> <p>Now, if we could “thicken” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a bit, namely to a <a class="existingWikiWord" href="/nlab/show/tubular+neighbourhood">tubular neighbourhood</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>↪</mo><mi>U</mi><mover><mo>↪</mo><mi>j</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex"> h \;\colon\; X \hookrightarrow U \stackrel{j}{\hookrightarrow} Y </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(X)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> without changing the K-theory 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>, then the element in question will just be the KK-element</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>!</mo><mo>∈</mo><mi>KK</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> j! \in KK(C_0(U), C(Y)) </annotation></semantics></math></div> <p>induced directly from the <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a> homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(U) \to C(Y)</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a> <a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, given by extending these functions by 0 to functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>. Or rather, it will be that element composed with the assumed KK-equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>KK</mi></msub></mrow></mover><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \psi \colon C(X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,. </annotation></semantics></math></div> <p>The bulk of the technical work in constructing the push-forward is in constructing this equivalence. (<a href="#BMRS07">BMRS 07, example 3.3</a>)</p> <p>In order for it to exist at all, assume that the <a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi><mo>≔</mo><msup><mi>h</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>T</mi><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> N_Y X \coloneqq h^\ast(T Y)/ T X </annotation></semantics></math></div> <p>has a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S(N_Y X)</annotation></semantics></math> for the associated <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>.</p> <p>Then there is an invertible element in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ι</mi> <mi>X</mi></msup><mo>!</mo><mo>∈</mo><msub><mi>KK</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \iota^X! \in KK_n(C(X), C_0(N_Y X)) </annotation></semantics></math></div> <p>hence a KK-equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ι</mi> <mi>X</mi></msup><mo>!</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota^X! \colon C(X) \stackrel{\simeq}{\to} C_0(N_Y X)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_0(-)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a> <a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a>.</p> <p>This is defined as follows. Consider the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>π</mi> <mi>n</mi> <mo>*</mo></msubsup><mi>S</mi><mo stretchy="false">(</mo><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\pi_n^\ast S(N_Y X) \to N_Y X</annotation></semantics></math> of this spinor to the normal bundle itself along the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>N</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\pi_N \colon N_Y X \to X</annotation></semantics></math>. Then…</p> <p>Moreover, a choice of a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> allows to find a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> between the <a class="existingWikiWord" href="/nlab/show/tubular+neighbourhood">tubular neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">U_{h(X)}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(X)</annotation></semantics></math> and a neighbourhood of the zero-section of of the normal bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>U</mi> <mrow><mi>h</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo>↪</mo><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Phi \colon U_{h(X)} \hookrightarrow N_Y X \,. </annotation></semantics></math></div> <p>This induces a KK-equivalence</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>Φ</mi><mo stretchy="false">]</mo><mo lspace="verythinmathspace">:</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>KK</mi></msub></mrow></mover><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Phi] \colon C_0(N_Y X) \stackrel{\simeq_{KK}}{\to} C_0(U) \,. </annotation></semantics></math></div> <p>Therefore the push-forward in operator K-theory along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>↪</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \hookrightarrow Y</annotation></semantics></math> is given by postcomposing in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>!</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>KK</mi></msub></mrow><mrow><msup><mi>i</mi> <mi>X</mi></msup><mo>!</mo></mrow></munderover><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>N</mi> <mi>Y</mi></msub><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>KK</mi></msub></mrow><mi>Φ</mi></munderover><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>j</mi><mo>!</mo></mrow></mover><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> h! \colon C(X) \underoverset{\simeq_{KK}}{i^X!}{\to} C_0(N_Y X) \underoverset{\simeq_{KK}}{\Phi}{\to} C_0(U) \stackrel{j!}{\to} C(Y) \,. </annotation></semantics></math></div> <h4 id="KKPushforwardAlongSubmersion">Along a proper submersion</h4> <p>(<a href="ConnesSkandalis84">Connes-Skandalis 84, above prop. 2.9</a>)</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">\pi \colon X \to Z</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/K-orientation">K-oriented</a> <a class="existingWikiWord" href="/nlab/show/proper+map">proper</a> <a class="existingWikiWord" href="/nlab/show/submersion">submersion</a> of compact smooth manifolds, the push-forward map along it is reduced to the <a href="#KKPushForwardAlongEmbedding">above</a> case of an embedding by</p> <ol> <li> <p>using that by the <a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a> every compact <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> may be embedded into some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2q}</annotation></semantics></math> such as to yield an embedding</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Z</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>q</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> h \colon X \to Z \times \mathbb{R}^{2 q} </annotation></semantics></math></div></li> <li> <p>using that there is a KK-equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ι</mi> <mi>Z</mi></msup><mo>!</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>KK</mi></msub></mrow></mover><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Z</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>2</mn><mi>q</mi></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \iota^Z! \colon C(Z) \stackrel{\simeq_{KK}}{\to} C_0(Z \times \mathbb{R}^{2q}) \,. </annotation></semantics></math></div></li> </ol> <p>The resulting push-forward is then given by postcomposition in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>!</mo><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>h</mi><mo>!</mo></mrow></mover><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Z</mi><mo>×</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mi>q</mi><mo stretchy="false">)</mo><munderover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>KK</mi></msub></mrow><mrow><mo stretchy="false">(</mo><msup><mi>ι</mi> <mi>Z</mi></msup><mo>!</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></munderover><mi>C</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi! \colon C(X) \stackrel{h!}{\to} C_0(Z \times \mathbb{R}^{2}q) \underoverset{\simeq_{KK}}{(\iota^Z!)^{-1}}{\to} C(Z) \,. </annotation></semantics></math></div> <p>(<a href="#BMRS07">BMRS 07, example 3.4</a>)</p> <h4 id="AlongAFibrationOfClosedSpinCManifolds">Along a smooth fibration of closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-manifolds</h4> <p>Specifically, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">\pi \colon X \to Z</annotation></semantics></math> a smooth fibration over a closed smooth manifold whose <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">X/Z</annotation></semantics></math> are</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a><a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth</a> <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c</a> <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> of even <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a></li> </ul> <p>the push-forward element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>!</mo><mo>∈</mo><mi>KK</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi! \in KK(C_0(X), C_0(Z))</annotation></semantics></math> is given by the <a class="existingWikiWord" href="/nlab/show/Fredholm+module">Fredholm</a>-<a class="existingWikiWord" href="/nlab/show/Hilbert+module">Hilbert module</a> obatined from the fiberwise <a class="existingWikiWord" href="/nlab/show/spin%5Ec+Dirac+operator">spin^c Dirac operator</a> acting on the fiberwise <a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>. (<a href="ConnesSkandalis84">Connes-Skandalis 84, proof of lemma 4.7</a>, <a href="BMRS07">BMRS 07, example 3.9</a>).</p> <p>In detail, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> T(X/Z) \hookrightarrow T X </annotation></semantics></math></div> <p>for the sub-bundle of the total <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> on the <a class="existingWikiWord" href="/nlab/show/vertical+vectors">vertical vectors</a> and choose a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi></mrow></msup></mrow><annotation encoding="application/x-tex">g^{X/Z}</annotation></semantics></math> on this bundle (hence a collection of Riemannian metric on the fibers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">X/Z</annotation></semantics></math> smoothly varying along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math>). Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">S_{X/Z}</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/spinor+bundle">spinor bundle</a>.</p> <p>A choice of horizontal complenet <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>≃</mo><msup><mi>T</mi> <mi>H</mi></msup><mi>X</mi><mo>⊕</mo><mi>T</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T X \simeq T^H X \oplus T(X/Z)</annotation></semantics></math> induces an <a class="existingWikiWord" href="/nlab/show/affine+connection">affine connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∇</mo> <mrow><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\nabla^{X/Z}</annotation></semantics></math>. This combined with the <a class="existingWikiWord" href="/nlab/show/symbol+map">symbol map</a>/Clifford multiplication of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T^\ast (X/Z)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi></mrow></msub></mrow><annotation encoding="application/x-tex">S_{X/Z}</annotation></semantics></math> induces a fiberwise <a class="existingWikiWord" href="/nlab/show/spin%5Ec+Dirac+operator">spin^c Dirac operator</a>, acting in each fiber on the <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi><mo>,</mo><msub><mi>S</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L^2(X/Z, S_{X/Z})</annotation></semantics></math>.</p> <p>This yields a <a class="existingWikiWord" href="/nlab/show/Fredholm+module">Fredholm</a>-<a class="existingWikiWord" href="/nlab/show/Hilbert+bimodule">Hilbert bimodule</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><msub><mi>D</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi></mrow></msub><mo>,</mo><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi><mo>,</mo><msub><mi>S</mi> <mrow><mi>X</mi><mo stretchy="false">/</mo><mi>Z</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (D_{X/Z}, L^2(X/Z, S_{X/Z})) </annotation></semantics></math></div> <p>which defines an element in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo>!</mo><mo>∈</mo><mi>KK</mi><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>C</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi ! \in KK(C_0(X), C_0(Z)) \,. </annotation></semantics></math></div> <p>Postcompositon with this is the push-forward map in K/KK-theory, equivalently the <a class="existingWikiWord" href="/nlab/show/index">index</a> map of the collection of Dirac operators.</p> <h4 id="#KKPushforwardAlongGeneralMap">Along a general K-oriented map</h4> <p>(<a href="ConnesSkandalis84">Connes-Skandalis 84, def. 2.1</a>)</p> <p>Now for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> an arbitray <a class="existingWikiWord" href="/nlab/show/K-orientation">K-oriented</a> smooth proper map, we may reduce push-forward along it to the above two cases by factoring it through its <a class="existingWikiWord" href="/nlab/show/graph+map">graph map</a>, followed by projection to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</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>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>→</mo><mrow><mi>graph</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mo>×</mo><mi>Y</mi><mover><mo>→</mo><mrow><msub><mi>p</mi> <mi>Y</mi></msub></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \stackrel{graph(f)}{\to} X \times Y \stackrel{p_Y}{\to} Y \,. </annotation></semantics></math></div> <p>Hence push-forward along such a general map is postcomposition in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>!</mo><mo>≔</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo>!</mo><mo>∘</mo><mi>graph</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>!</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f! \coloneqq p_Y !\circ graph(f)! \,. </annotation></semantics></math></div> <p>(<a href="#BMRS07">BMRS 07, example 3.5</a>)</p> <h4 id="KKPushforwardAlongGeneralMap">In twisted K-theory</h4> <p>We discuss push forward in K-theory more generally by <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality+C%2A-algebras">Poincaré duality C*-algebras</a> hence <a class="existingWikiWord" href="/nlab/show/dual+objects">dual objects</a> in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>Q</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon Q \to X</annotation></semantics></math> be a map of <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msup><mi>B</mi> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\chi \colon X \to B^2 U(1)</annotation></semantics></math> modulate a <a class="existingWikiWord" href="/nlab/show/circle+2-bundle">circle 2-bundle</a> regarded as a <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twist for K-theory</a>. Then forming <a class="existingWikiWord" href="/nlab/show/twisted+groupoid+convolution+algebras">twisted groupoid convolution algebras</a> yields a <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> morphism of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>χ</mi></mrow></msub><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mover><mo>⟵</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><msub><mi>C</mi> <mi>χ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> C_{i^\ast \chi}(Q) \stackrel{i^\ast}{\longleftarrow} C_{\chi}(X) \,, </annotation></semantics></math></div> <p>with notation as in <a href="Poincar&#233;+duality+algebra#CStarAlgebraOf2BundleOnManifold">this definition</a>. By <a href="Poincar&#233;+duality+algebra#DualOfCompactManifoldWithTwist">this proposition</a> the <a class="existingWikiWord" href="/nlab/show/dual+morphism">dual morphism</a> is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mfrac><mrow><msub><mi>W</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>Q</mi></msub><mo stretchy="false">)</mo></mrow><mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>χ</mi></mrow></mfrac></msub><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mo>!</mo></msub></mrow></mover><msub><mi>C</mi> <mfrac><mrow><msub><mi>W</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><mi>χ</mi></mfrac></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{W_3(\tau_X)}{\chi}}(X) \,. </annotation></semantics></math></div> <p>If we assume that <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> has a <a class="existingWikiWord" href="/nlab/show/spin%5Ec+structure">spin^c structure</a> then this is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mfrac><mrow><msub><mi>W</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>Q</mi></msub><mo stretchy="false">)</mo></mrow><mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>χ</mi></mrow></mfrac></msub><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mo>!</mo></msub></mrow></mover><msub><mi>C</mi> <mfrac><mn>1</mn><mi>χ</mi></mfrac></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_{\frac{W_3(\tau_Q)}{i^\ast \chi}}(Q) \stackrel{i_!}{\longrightarrow} C_{\frac{1}{\chi}}(X) \,. </annotation></semantics></math></div> <p>Postcomposition with this map in <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> now yields a map from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msub><mi>W</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>Q</mi></msub><mo stretchy="false">)</mo></mrow><mrow><msup><mi>i</mi> <mo>*</mo></msup><mi>χ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{W_3(\tau_Q)}{i^\ast \chi}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>χ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\chi^{-1}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mo>!</mo></msub><mo lspace="verythinmathspace">:</mo><msub><mi>K</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>W</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>Q</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>i</mi> <mo>*</mo></msup><mi>χ</mi></mrow></msub><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>K</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>χ</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> i_! \colon K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) \to K_{\bullet -\chi} \,. </annotation></semantics></math></div> <p>If we here think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>Q</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon Q \hookrightarrow X</annotation></semantics></math> as being the inclusion of a <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi></mrow><annotation encoding="application/x-tex">\chi</annotation></semantics></math> would be the class of the <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background</a> <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> and an element</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>ξ</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>K</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><msub><mi>W</mi> <mn>3</mn></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>Q</mi></msub><mo stretchy="false">)</mo><mo>−</mo><msup><mi>i</mi> <mo>*</mo></msup><mi>χ</mi></mrow></msub><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [\xi] \in K_{\bullet + W_3(\tau_Q) - i^\ast \chi}(Q) </annotation></semantics></math></div> <p>is called (the K-class of) a <em><a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton gauge field</a></em> on the D-brane satisfying the <em><a class="existingWikiWord" href="/nlab/show/Freed-Witten-Kapustin+anomaly+cancellation">Freed-Witten-Kapustin anomaly cancellation</a></em> mechanism. (The orginal <em><a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly+cancellation">Freed-Witten anomaly cancellation</a></em> assumes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi></mrow><annotation encoding="application/x-tex">\xi</annotation></semantics></math> given by a <a class="existingWikiWord" href="/nlab/show/twisted+unitary+bundle">twisted line bundle</a> in which case it exhibits a <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>.) Finally its <a class="existingWikiWord" href="/nlab/show/fiber+integration">push-forward</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><msub><mi>i</mi> <mo>!</mo></msub><mi>ξ</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mi>K</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>χ</mi></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [i_! \xi] \in K_{\bullet- \chi}(X) </annotation></semantics></math></div> <p>is called the corresponding <em><a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a></em>.</p> <h3 id="of_cohesive_differential_form_data">Of cohesive differential form data</h3> <p>In <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> realized in <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a> there is a canonical fiber integration map for the curvature coefficients of a given diffential cohomology theory. See at <em><a href="integration+of+differential+forms#InCohesiveHomotopyTypeTheory">integration of differential forms – In cohesive homotopy-type theory</a></em>.</p> <h2 id="examples">Examples</h2> <h3 id="to_the_point">To the point</h3> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a point, one obtains integration aginst the <a class="existingWikiWord" href="/nlab/show/fundamental+class">fundamental class</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>E</mi></msub><mo>:</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>dim</mi><mi>E</mi></mrow></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \int_E:H^\bullet(E)\to H^{\bullet-dim E}(*) </annotation></semantics></math></div> <p>taking values in the coefficients of the given cohomology theory. Note that in this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mi>n</mi></msup><msub><mi>B</mi> <mo>+</mo></msub><mo>=</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Sigma^n B_+=S^n</annotation></semantics></math>, and this hints to a relationship between the Thom-Pontryagin construction and <a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a>. And indeed <a class="existingWikiWord" href="/nlab/show/Atiyah+duality">Atiyah duality</a> gives a homotopy equivalence between the <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a> of the stable normal bundle of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and the Spanier-Whitehead dual of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. …</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck-Riemann-Roch+theorem">Grothendieck-Riemann-Roch theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transfer+context">transfer context</a>, <a class="existingWikiWord" href="/nlab/show/sheaf+with+transfer">sheaf with transfer</a>, <a class="existingWikiWord" href="/nlab/show/Becker-Gottlieb+transfer">Becker-Gottlieb transfer</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+by+push-forward">geometric quantization by push-forward</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+Umkehr+map">twisted Umkehr map</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+K-theory">fiber integration in K-theory</a></li> </ul> </li> </ul> <div> <p>The following terms all refer to essentially the same concept:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+generalized+cohomology">fiber integration in generalized cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/push-forward+in+generalized+cohomology">push-forward in generalized cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Umkehr+map">Umkehr map</a>, <a class="existingWikiWord" href="/nlab/show/twisted+Umkehr+map">twisted Umkehr map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gysin+map">Gysin map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+collapse+map">Pontrjagin-Thom collapse map</a></p> </li> </ul></div> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li id="Adams74"> <p><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, pages 25-27 in <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalized+homology">Stable homotopy and generalized homology</a></em>, Chicago Lectures in mathematics, 1974</p> </li> <li id="Kochmann96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochmann">Stanley Kochmann</a>, section 4.3 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</p> </li> </ul> <p>Fiber integration of differential forms is discussed in section VII of volume I of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Werner+Greub">Werner Greub</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Halperin">Stephen Halperin</a>, <a class="existingWikiWord" href="/nlab/show/Ray+Vanstone">Ray Vanstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Connections%2C+Curvature%2C+and+Cohomology">Connections, Curvature, and Cohomology</a></em> Academic Press (1973)</li> </ul> <p>A quick summary can be found from <a href="http://www.math.wisc.edu/~gstgc/slides/Koytcheff.pdf#page=14">slide 14</a> on in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Robin+Koytcheff">Robin Koytcheff</a>, <em>A homotopy-theoretic view of Bott-Taubes integrals</em> (<a href="http://www.math.wisc.edu/~gstgc/slides/Koytcheff.pdf">pdf slides</a>)</li> </ul> <p>More details are in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <a class="existingWikiWord" href="/nlab/show/John+Klein">John Klein</a>, <em>Umkehr Maps</em> (<a href="http://arxiv.org/abs/0711.0540">arXiv:0711.0540</a>)</li> </ul> <h3 id="ReferencesInKKTheory">In noncommutative topology and KK-theory</h3> <p>Push-forward in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> is discussed in</p> <ul id="CareyWang05"> <li><a class="existingWikiWord" href="/nlab/show/Alan+Carey">Alan Carey</a>, <a class="existingWikiWord" href="/nlab/show/Bai-Ling+Wang">Bai-Ling Wang</a>, <em>Thom isomorphism and Push-forward map in twisted K-theory</em> (<a href="http://arxiv.org/abs/math/0507414">arXiv:math/0507414</a>)</li> </ul> <p>and section 10 of (<a href="#ABG10">ABG, 10</a>)</p> <p>Discussion of fiber integration <a class="existingWikiWord" href="/nlab/show/Gysin+maps">Gysin maps</a>/Umkehr maps in <a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a>/<a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a> as <a href="#InKKTheory">above</a> is in the following references.</p> <p>The definition of the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>!</mo><mo>∈</mo><mi>KK</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>C</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f! \in KK(C(X), C(Y))</annotation></semantics></math> for a <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>-oriented map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> between smooth manifolds goes back to section 11 in</p> <ul> <li id="Connes82"><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <em>A survey of foliations and operator algebras</em>, Proceedings of the A.M.S., 38, 521-628 (1982) (<a href="http://www.alainconnes.org/docs/foliationsfine.pdf">pdf</a>)</li> </ul> <p>The functoriality of this construction is demonstrated in section 2 of the following article, which moreover generalizes the construction to maps between <a class="existingWikiWord" href="/nlab/show/foliations">foliations</a> hence to KK-elements between <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebras">groupoid convolution algebras</a> of <a class="existingWikiWord" href="/nlab/show/holonomy+groupoids">holonomy groupoids</a>:</p> <ul> <li id="ConnesSkandalis84"><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Georges+Skandalis">Georges Skandalis</a>, <em>The longitudinal index theorem for foliations</em>. Publ. Res. Inst. Math. Sci. 20, <p>no. 6, 1139–1183 (1984) (<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.165.4218">web</a>)</p> </li> </ul> <p>More on this is in</p> <ul> <li id="HilsumSkandalis87"><a class="existingWikiWord" href="/nlab/show/Michel+Hilsum">Michel Hilsum</a>, <a class="existingWikiWord" href="/nlab/show/Georges+Skandalis">Georges Skandalis</a>, <em>Morphismes K-orienté d’espace de feuille et fonctoralité en théorie de Kasparov</em>, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 20 no. 3 (1987), p. 325-390 (<a href="http://www.numdam.org/item?id=ASENS_1987_4_20_3_325_0">numdam</a>)</li> </ul> <p>(the article that introuced <a class="existingWikiWord" href="/nlab/show/Hilsum-Skandalis+morphisms">Hilsum-Skandalis morphisms</a>).</p> <p>This is further generalized to <a class="existingWikiWord" href="/nlab/show/circle+2-bundle">circle 2-bundle</a>-twisted convolution algebras of foliations in</p> <ul> <li id="RouseWang10">Paulo Carrillo Rouse, <a class="existingWikiWord" href="/nlab/show/Bai-Ling+Wang">Bai-Ling Wang</a>, <em>Twisted longitudinal index theorem for foliations and wrong way functoriality</em> (<a href="http://arxiv.org/abs/1005.3842">arXiv:1005.3842</a>)</li> </ul> <p>Dicussion for general <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> is in section 3 of</p> <ul> <li id="BMRS07">Jacek Brodzki, <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>Noncommutative correspondences, duality and D-branes in bivariant K-theory</em>, Adv. Theor. Math. Phys.13:497-552,2009 (<a href="http://arxiv.org/abs/0708.2648">arXiv:0708.2648</a>)</li> </ul> <p>and specifically including also <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> again (and the relation to <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a>) in section 7 of</p> <ul> <li id="BrodzkiMathaiRosenbergSzabo06">Jacek Brodzki, <a class="existingWikiWord" href="/nlab/show/Varghese+Mathai">Varghese Mathai</a>, <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>D-Branes, RR-Fields and Duality on Noncommutative Manifolds</em>, Commun. Math. Phys. 277:643-706,2008 (<a href="http://arxiv.org/abs/hep-th/0607020">arXiv:hep-th/0607020</a>)</li> </ul> <h3 id="abstract_formulation">Abstract formulation</h3> <p>The abstract formulation in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> via <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-module+bundles">(infinity,1)-module bundles</a> is sketched in section 9 of</p> <ul> <li id="ABG10"><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <em>Twists of K-theory and TMF</em>, in Robert S. Doran, Greg Friedman, <a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <em>Superstrings, Geometry, Topology, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">C^*</annotation></semantics></math>-algebras</em>, Proceedings of Symposia in Pure Mathematics <a href="http://www.ams.org/bookstore-getitem/item=PSPUM-81">vol 81</a>, American Mathematical Society (<a href="http://arxiv.org/abs/1002.3004">arXiv:1002.3004</a>)</li> </ul> <p>and in section 10 of</p> <ul> <li id="ABG11"><a class="existingWikiWord" href="/nlab/show/Matthew+Ando">Matthew Ando</a>, <a class="existingWikiWord" href="/nlab/show/Andrew+Blumberg">Andrew Blumberg</a>, <a class="existingWikiWord" href="/nlab/show/David+Gepner">David Gepner</a>, <em>Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map</em> (<a href="http://arxiv.org/abs/1112.2203">arXiv:1112.2203</a>)</li> </ul> <p>This is reviewed and used also in</p> <ul> <li id="Nuiten13"><a class="existingWikiWord" href="/nlab/show/Joost+Nuiten">Joost Nuiten</a>, <em><a class="existingWikiWord" href="/schreiber/show/master+thesis+Nuiten">Cohomological quantization of boundary prequantum field theory</a></em>, MSc thesis, Utrecht 2013</li> </ul> <p>Formulation of this in <a class="existingWikiWord" href="/nlab/show/linear+homotopy-type+theory">linear homotopy-type theory</a> is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Homotopy-type+semantics+for+quantization">Homotopy-type semantics for quantization</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 19, 2024 at 09:31:03. 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