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href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <ul> <li><a href='#the_kgroup'>The K-group</a></li> <li><a href='#ReducedKGroup'>The reduced K-group</a></li> <li><a href='#TheRelativeKGroup'>The relative K-group</a></li> <li><a href='#TheGradedKGroups'>The graded K-groups</a></li> <li><a href='#ForNonCompactSpaces'>For non-compact spaces</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#homotopy_invariance'>Homotopy invariance</a></li> <li><a href='#ExactSequences'>Exact sequences</a></li> <li><a href='#ExternalProducts'>External product</a></li> <li><a href='#fundamental_product_theorem'>Fundamental product theorem</a></li> <li><a href='#BottPeriodicities'>Bott periodicity</a></li> <li><a href='#GradedRingStructure'>Graded-commutative ring structure</a></li> <li><a href='#ClassifyingSpace'>Classifying space</a></li> <li><a href='#of_noncompact_spaces'>Of non-compact spaces</a></li> <li><a href='#AsAGeneralizedCohomologyTheory'>As a generalized cohomology theory</a></li> <li><a href='#ComplexOrientationAndFormalGroupLaw'>Complex orientation and formal group law</a></li> <li><a href='#spectrum'>Spectrum</a></li> <li><a href='#ring_spectrum'>Ring spectrum</a></li> <li><a href='#chromatic_filtration'>Chromatic filtration</a></li> <li><a href='#as_the_shape_of_the_smooth_ktheory_spectrum'>As the shape of the smooth K-theory spectrum</a></li> <li><a href='#RelationToAlgebraicKTheory'>Relation to algebraic K-theory</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='#for_noncompact_spaces_2'>For non-compact spaces</a></li> <li><a href='#dbrane_charge'>D-brane charge</a></li> <li><a href='#topological_phases_of_matter'>Topological phases of matter</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>What is called <em>topological</em> <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> is a collection of <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theories whose cocycles in degree 0 on a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> may be represented by pairs of <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a>, real or complex ones, 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> modulo a certain equivalence relation.</p> <p>The following is the quick idea. For a detailed introduction see <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topological+K-Theory">Introduction to Topological K-Theory</a></em>.</p> <p>First, recall that for <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> a <a class="existingWikiWord" href="/nlab/show/field">field</a> then 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>-<a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> over a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">V \to X</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> are <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> which vary over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in a controlled way. Explicitly this means that there exits an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <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> (the <em><a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank</a></em> of the vector bundle) and a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo>→</mo><mi>V</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">U_i \times k^n \to V|_{U_i}</annotation></semantics></math> over <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> which is fiberwise 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>-<a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a>.</p> <p>Vector bundles are of central interest in large parts of <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> and <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, for instance in <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> and <a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>. But the collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Vect(X)_{/\sim}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of vector bundles over a given space is in general hard to analyze. One reason for this is that these are classified in degree-1 <em><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></em> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the (<a class="existingWikiWord" href="/nlab/show/nonabelian+group">nonabelian</a>) <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,k)</annotation></semantics></math>. K-theory may roughly be thought of as the result of forcing vector bundles to be classified by an abelian <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a>.</p> <p>To that end, observe that all natural operations on <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> generalize to vector bundles by applying them <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>-wise. Notably there is the fiberwise <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a>, also called the <em><a class="existingWikiWord" href="/nlab/show/Whitney+sum">Whitney sum</a></em> operation. This operation gives the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Vect(X)_{/\sim}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of vector bundles the structure of an <a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a> (<a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>,</mo><mo>⊕</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Vect(X)_{/\sim},\oplus)</annotation></semantics></math>.</p> <p>Now as under direct sum the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of vector spaces adds, similarly under <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> their <a class="existingWikiWord" href="/nlab/show/rank">rank</a> adds. Hence in analogy to how one passes from the additive <a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a> (<a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>) of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> to the additive <a class="existingWikiWord" href="/nlab/show/group">group</a> of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> by adjoining formal additive inverses, so one may adjoin formal additive inverses to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>,</mo><mo>⊕</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Vect(X)_{/\sim},\oplus)</annotation></semantics></math>. By a general prescription (“<a class="existingWikiWord" href="/nlab/show/Grothendieck+group+of+a+commutative+monoid">Grothendieck group of a commutative monoid</a>”) this is achieved by first passing to the larger class of <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>−</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V_+,V_-)</annotation></semantics></math> of vector bundles (“<a class="existingWikiWord" href="/nlab/show/virtual+vector+bundles">virtual vector bundles</a>”), and then <a class="existingWikiWord" href="/nlab/show/quotient">quotienting</a> out the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> given by</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>V</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>∼</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>+</mo></msub><mo>⊕</mo><mi>W</mi><mo>,</mo><msub><mi>V</mi> <mo>−</mo></msub><mo>⊕</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (V_+, V_-) \sim (V_+ \oplus W , V_- \oplus W) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>∈</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding="application/x-tex">W \in Vect(X)_{/\sim}</annotation></semantics></math>. The resulting set of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> with group operation given on representatives by</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>V</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo>⊕</mo><mo stretchy="false">[</mo><mi>V</mi><msub><mo>′</mo> <mo>+</mo></msub><mo>,</mo><mi>V</mi><msub><mo>′</mo> <mo>−</mo></msub><mo stretchy="false">]</mo><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mo>+</mo></msub><mo>⊕</mo><mi>V</mi><msub><mo>′</mo> <mo>+</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>−</mo></msub><mo>⊕</mo><mi>V</mi><msub><mo>′</mo> <mo>−</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [V_+, V_-] \oplus [V'_+, V'_-] \coloneqq (V_+ \oplus V'_+, V_- \oplus V'_-) </annotation></semantics></math></div> <p>and with the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>V</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>−</mo></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[V_+,V_-]</annotation></semantics></math> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">[</mo><msub><mi>V</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>V</mi> <mo>−</mo></msub><mo>,</mo><msub><mi>V</mi> <mo>+</mo></msub><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> -[V_+, V_-] = [V_-, V_+] \,. </annotation></semantics></math></div> <p>This <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> obtained from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>,</mo><mo>⊕</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Vect(X)_{/\sim}, \oplus)</annotation></semantics></math> is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math> and often called <em>the K-theory</em> of the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Here the letter “K” (due to <a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>) originates as a shorthand for the German word <em>Klasse</em>, referring to the above process of forming <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of (<a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of) vector bundles.</p> <p>This simple construction turns out to yield remarkably useful groups of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> <a class="existingWikiWord" href="/nlab/show/invariants">invariants</a>. A variety of deep facts in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> have fairly elementary proofs in terms of topological K-theory, for instance the <a class="existingWikiWord" href="/nlab/show/Hopf+invariant+one">Hopf invariant one</a> problem (<a href="#AdamsAtiyah66">Adams-Atiyah 66</a>).</p> <p>One defines the “higher” K-groups of a topological space to be those of its higher <a class="existingWikiWord" href="/nlab/show/reduced+suspensions">reduced suspensions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>K</mi><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mi>n</mi></msup><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K^{-n}(X) = K(\Sigma^n X) \,. </annotation></semantics></math></div> <p>The assignment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><msup><mi>K</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \mapsto K^\bullet(X)</annotation></semantics></math> turns out to share many properties of the assignment of <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> groups <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \mapsto H^n(X,\mathbb{Z})</annotation></semantics></math>. One says that topological K-theory is a <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theory. As such it is <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">represented</a> by a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{C}</annotation></semantics></math> this is called <a class="existingWikiWord" href="/nlab/show/KU">KU</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{R}</annotation></semantics></math> this is called <a class="existingWikiWord" href="/nlab/show/KO">KO</a>. (There is also the unification of both in <a class="existingWikiWord" href="/nlab/show/KR">KR</a>-theory.)</p> <p>One of the basic facts about topological K-theory, rather unexpected from the definition, is that these higher K-groups repeat <em>periodically</em> in the degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{R}</annotation></semantics></math> the periodicity is 8, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">k = \mathbb{C}</annotation></semantics></math> it is 2. This is called <em><a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a></em>.</p> <p>It turns out that an important source of <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundles">virtual vector bundles</a> representing classes in <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> are <a class="existingWikiWord" href="/nlab/show/index+bundles">index bundles</a>: Given a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian</a> <a class="existingWikiWord" href="/nlab/show/spin+structure">spin</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <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>, then there is a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">S \to B</annotation></semantics></math> called the <em><a class="existingWikiWord" href="/nlab/show/spin+bundle">spin bundle</a></em> 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>, which carries a <a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a>, called the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</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>. The <a class="existingWikiWord" href="/nlab/show/index+of+a+Dirac+operator">index of a Dirac operator</a> is the formal difference of its <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> by its <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ker</mi><mi>D</mi><mo>,</mo><mi>coker</mi><mi>D</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[ker D, coker D]</annotation></semantics></math>. Now given a continuous family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">D_x</annotation></semantics></math> of Dirac operators/Fredholm operators, parameterized by some topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then these indices combine to a class in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math>.</p> <p>It is via this construction that topological K-theory connects to <a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a> (see e.g. <a class="existingWikiWord" href="/nlab/show/Karoubi+K-theory">Karoubi K-theory</a>) and <a class="existingWikiWord" href="/nlab/show/index+theory">index theory</a>.</p> <p>As the terminology indicates, both <a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a> and <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a> originate in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. Accordingly, K-theory plays a central role in various areas of <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, for instance in the theory of <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> (“<a class="existingWikiWord" href="/nlab/show/spin%5Ec+quantization">spin^c quantization</a>”) in the theory of <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> (where it models <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> and <a class="existingWikiWord" href="/nlab/show/RR-fields">RR-fields</a>) and in the theory of <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> via <a class="existingWikiWord" href="/nlab/show/spectral+triples">spectral triples</a> (see below).</p> <p>All these geometric constructions have an <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebraic</a> incarnation: by the topological <a class="existingWikiWord" href="/nlab/show/Serre-Swan+theorem">Serre-Swan theorem</a> then <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> of finite rank are equivalently <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over the <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> on the base space. Using this relation one may express K-theory classes entirely operator algebraically, this is called <em><a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a></em>. Now <a class="existingWikiWord" href="/nlab/show/Dirac+operators">Dirac operators</a> are generalized to <a class="existingWikiWord" href="/nlab/show/Fredholm+operators">Fredholm operators</a>.</p> <p>There are more <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> than arising as <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a> of <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, namely non-commutative C<em>-algebras. One may think of these as defining <a class="existingWikiWord" href="/nlab/show/non-commutative+geometry">non-commutative geometry</a>, but the definition of <a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a> immediately generalizes to this situation (see also at <em><a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></em>).</em></p> <p>While the <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a> of a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian</a> <a class="existingWikiWord" href="/nlab/show/spin+structure">spin</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> remembers only the underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, one may algebraically encode the <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a> and <a class="existingWikiWord" href="/nlab/show/Riemannian+structure">Riemannian structure</a> by passing from <a class="existingWikiWord" href="/nlab/show/Fredholm+modules">Fredholm modules</a> to “<a class="existingWikiWord" href="/nlab/show/spectral+triples">spectral triples</a>”. This may for instance be used to algebraically encode the spin physics underlying the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> and <a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a> plays a crucial role in this.</p> <h2 id="Definition">Definition</h2> <p>The following discussion of topological K-theory in terms of <a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>. For more abstract perspectives see for instance <em><a class="existingWikiWord" href="/nlab/show/Snaith%27s+theorem">Snaith's theorem</a></em> and other pointers at <em><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></em>.</p> <p>Assumed background for the following is the content of</p> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/Grothendieck+group+of+a+commutative+monoid">Grothendieck group of a commutative monoid</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a></em></p> </li> </ul> <p>Throughout, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+field">topological field</a>, usually the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> or the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>.</p> <p>In the following we take</p> <ol> <li> <p><em><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a></em> to mean <em><a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional vector space</a></em> over <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> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a></em> to mean <em><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> over <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> of finite <a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank</a></em>.</p> </li> </ol> <p>We say <em><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></em> for <em><a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a> with <a class="existingWikiWord" href="/nlab/show/unitality">unit</a></em>.</p> <p>For the most part below we will invoke the assumption that the base <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>. Because then the following statement holds, which is crucial in some places:</p> <div class="num_lemma" id="DirectSumHasInverseUpToTrivialBundle"> <h6 id="lemma">Lemma</h6> <p><strong>(over a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> every <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> is a <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct summand</a> of a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>)</strong></p> <p>For every <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> there exists a <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\tilde E \to X</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</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><msub><mo>⊕</mo> <mi>X</mi></msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover><mo>≃</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> E \oplus_X \tilde E \simeq X \times \mathbb{R}^{n} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>.</p> </div> <p>For <strong>proof</strong> see <a href="topological+vector+bundle#TopologicalVectorbundleOverCompactHausdorffSpaceIsDirectSummandOfTrivialBundle">this prop.</a> at <em><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a></em>.</p> <h3 id="the_kgroup">The K-group</h3> <p>The starting point is the simple observation that the operation of <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> yields a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> structure (<a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a> with <a class="existingWikiWord" href="/nlab/show/unitality">unit</a>) on <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a>, which however is lacking <a class="existingWikiWord" href="/nlab/show/inverse+elements">inverse elements</a> and hence is not an actual <a class="existingWikiWord" href="/nlab/show/group">group</a>.</p> <div class="num_defn" id="SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</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>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Vect(X)_{/\sim}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. The operation of <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⊕</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (-)\oplus_X (-) \;\colon\; Vect(X) \times Vect(X) \longrightarrow Vect(X) </annotation></semantics></math></div> <p>descends to this <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> by <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</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>E</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo>⊕</mo> <mi>X</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [E_1] + [E_2] \;\coloneqq\; [E_1 \oplus_X E_2] </annotation></semantics></math></div> <p>to yield the structure of a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (<a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a> with <a class="existingWikiWord" href="/nlab/show/unitality">unit</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( Vect(X)_{/\sim}, + \right) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The operation of <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> on <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> in def. <a class="maruku-ref" href="#SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX"></a> is indeed not a <a class="existingWikiWord" href="/nlab/show/group">group</a>:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> be a chosen point 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> and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>rk</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>⟶</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> rk_x \;\colon\; Vect(X)_{/\sim} \longrightarrow \mathbb{N} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/function">function</a> which takes a <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> to the <a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank</a> over the <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> of the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> <p>Then under <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> the rank is additive</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo>⊕</mo> <mi>X</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> rk_x(E_1 \oplus_X E_2) \,=\, rk_x(E_1) + rk_x(E_1) \,. </annotation></semantics></math></div> <p>Now since the <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> under <a class="existingWikiWord" href="/nlab/show/addition">addition</a> are just a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (<a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a> with <a class="existingWikiWord" href="/nlab/show/unitality">unit</a>), with no element except zero having an <a class="existingWikiWord" href="/nlab/show/inverse+element">inverse element</a> under the additive operation, it follows immediately that a necessary condition for the <a class="existingWikiWord" href="/nlab/show/isomorphism+class">isomorphism class</a> of a <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> to be invertible under <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> is that its <a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank of a vector bundle</a> be zero. But there is only one such class of vector bundles, in fact there is only one such vector bundle, namely the unique rank-zero bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">X \times k^0</annotation></semantics></math>, necessarily a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>.</p> <p>Now for the <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a> <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><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{N},+)</annotation></semantics></math> it is a time honored fact that it is interesting and useful to rectify its failure of being a <a class="existingWikiWord" href="/nlab/show/group">group</a> by <a class="existingWikiWord" href="/nlab/show/universal+construction">universally</a> forcing it to become one. This is a process called <em><a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a></em> and the group completion of the natural numbers is the additive group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbb{Z},+)</annotation></semantics></math>.</p> <p>The idea is hence to apply group completion also to the monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Vect(X)_{/\sim}, +)</annotation></semantics></math>, and so that the <a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank</a> operation above becomes a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>.</p> </div> <p>An explicit construction of <a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a> of a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> is called the <em><a class="existingWikiWord" href="/nlab/show/Grothendieck+group+of+a+commutative+monoid">Grothendieck group of a commutative monoid</a></em>.</p> <div class="num_defn" id="KGroupByGrothendieckGroup"> <h6 id="definition_3">Definition</h6> <p><strong>(K-group as the <a class="existingWikiWord" href="/nlab/show/Grothendieck+group+of+a+commutative+monoid">Grothendieck group</a> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(X) \;\coloneqq\; K(Vect(X)_{/\sim}, +) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/Grothendieck+group+of+a+commutative+monoid">Grothendieck group</a> of the <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> (abelian <a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a> with <a class="existingWikiWord" href="/nlab/show/unitality">unit</a>) of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</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> from def. <a class="maruku-ref" href="#SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX"></a>.</p> <p>This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/group">group</a> whose elements are <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of pairs</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><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>+</mo></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>×</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> ([E_+], [E_-]) \; \in Vect(X)_{/\sim} \times Vect(X)_{/\sim} </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</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>, with respect to the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></p> <div class="maruku-equation" id="eq:DefiningEquivalenceRelation"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>+</mo></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>∼</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>F</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>F</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><munder><mo>∃</mo><mrow><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow></munder><mrow><mo>(</mo><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>+</mo></msub><msub><mo>⊕</mo> <mi>X</mi></msub><mi>G</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>−</mo></msub><msub><mo>⊕</mo> <mi>X</mi></msub><mi>G</mi><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>F</mi> <mo>+</mo></msub><msub><mo>⊕</mo> <mi>X</mi></msub><mi>H</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>F</mi> <mo>−</mo></msub><msub><mo>⊕</mo> <mi>X</mi></msub><mi>H</mi><mo stretchy="false">]</mo><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \big( \left( [E_+], [E_-] \right) \;\sim\; \left( [F_+, F_-] \right) \big) \;\Leftrightarrow\; \left( \underset{[G],[H] \in Vect(X)_{/\sim}}{\exists} \left( \left([E_+ \oplus_X G] , [E_- \oplus_X G]\right) \,=\, \left([F_+ \oplus_X H] , [F_- \oplus_X H]\right) \right) \right) \,. </annotation></semantics></math></div> <p>Here a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>+</mo></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">([E_+], [E_-])</annotation></semantics></math> is also called a <em><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></em>, and its equivalence class under the above equivalence relation is also denoted</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>E</mi> <mo>+</mo></msub><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [E_+] - [E_-] \;\;\in K(X) \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a>, hence equipped with a choice of point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> then the difference of <a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">ranks</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">rk_x(-)</annotation></semantics></math> of the representing vector bundles over the <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in 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>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>+</mo></msub><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> rk_x( [E_+] - [E_-] ) \;\coloneqq\; rk_x(E_+) - rk_x(E_-) \in \mathbb{Z} </annotation></semantics></math></div> <p>is called the <em>virtual rank</em> of the virtual vector bundle.</p> </div> <div class="num_example" id="KGroupOfThePoint"> <h6 id="example">Example</h6> <p><strong>(K-group of the point is the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>)</strong></p> <p>Let <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 = \ast</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/point">point</a>. Then a <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector 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> is just a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> Vect(\ast) \simeq Vect </annotation></semantics></math></div> <p>and an isomorphism of vector bundles is just a bijective <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a>.</p> <p>Since <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> are <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> precisely if they have the same <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>, the <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (<a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a> with <a class="existingWikiWord" href="/nlab/show/unitality">unit</a>) of isomorphism classes of vector bundles over the point (def. <a class="maruku-ref" href="#SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX"></a>) is the <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mo>*</mo><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mi>ℕ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( Vect(\ast)_{/\sim}, + \right) \;\simeq\; \left( \mathbb{N}, + \right) \,. </annotation></semantics></math></div> <p>Accordingly the K-group of the point is the <a class="existingWikiWord" href="/nlab/show/Grothendieck+group+of+a+commutative+monoid">Grothendieck group</a> of the <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>, which is the additive group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> (<a href="Grothendieck+group+of+a+commutative+monoid#GrothendieckGroupOfNaturalNumbersUnderAdditionIsTheIntegers">this example</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(\ast) \simeq (\mathbb{Z}, +) </annotation></semantics></math></div> <p>and this identification is the assignment of <em>virtual rank</em> (def. <a class="maruku-ref" href="#KGroupByGrothendieckGroup"></a>).</p> </div> <div class="num_prop" id="OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle"> <h6 id="proposition">Proposition</h6> <p><strong>(on <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a> all <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundles">virtual vector bundles</a> are formal difference by a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>, then every <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector 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> (def. <a class="maruku-ref" href="#KGroupByGrothendieckGroup"></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><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [E] - [X \times k^n] </annotation></semantics></math></div> <p>(i.e. with negative component represented by a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>).</p> </div> <div class="proof"> <h6 id="proof">Proof</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> compact Hausdorff then lemma <a class="maruku-ref" href="#DirectSumHasInverseUpToTrivialBundle"></a> implies that for every <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">E_-</annotation></semantics></math> there exists a topological vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>−</mo></msub></mrow><annotation encoding="application/x-tex">\tilde E_-</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>−</mo></msub><msub><mo>⊕</mo> <mi>X</mi></msub><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>−</mo></msub><mo>≃</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">E_- \oplus_X \tilde E_- \simeq X \times k^n</annotation></semantics></math>, and hence</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>E</mi> <mo>+</mo></msub><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo>=</mo><munder><munder><mrow><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>+</mo></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>−</mo></msub><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder><mrow><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo></mrow></munder><mo>−</mo><munder><munder><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>−</mo></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mo>⏟</mo></munder><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">]</mo></mrow></munder><mo>=</mo><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [E_+] - [E_-] = \underset{[E]}{\underbrace{[E_+] + [\tilde E_-]}} - \underset{[X \times k^n]}{\underbrace{ \left( [E_-] + [\tilde E_-] \right) } } = [E] - [X \times k^n] \,. </annotation></semantics></math></div></div> <div class="num_remark" id="KTheoryRing"> <h6 id="remark_2">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math> from <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product of vector bundles</a>)</strong></p> <p>Also the operation of <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product of vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> descends to <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a> and makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mo>∼</mo></msub><mo>,</mo><mo>⊕</mo><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Vect(X)_{\sim}, \oplus, \otimes )</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/semi-ring">semi-ring</a> (<a class="existingWikiWord" href="/nlab/show/rig">rig</a>).</p> <p>(This is the shadow under passing to isomorphism classes of the fact that the <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Vect(X)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/distributive+monoidal+category">distributive monoidal category</a> under <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product of vector bundles</a>.)</p> <p>This multiplicative structure passes to the K-group (def. <a class="maruku-ref" href="#KGroupByGrothendieckGroup"></a>) by the formula</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>E</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>E</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mo>⋅</mo><mo stretchy="false">[</mo><msub><mi>F</mi> <mo>+</mo></msub><mo>,</mo><msub><mi>F</mi> <mo>−</mo></msub><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>+</mo></msub><msub><mo>⊗</mo> <mi>X</mi></msub><msub><mi>F</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><msub><mo>⊕</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>−</mo></msub><msub><mo>⊗</mo> <mi>X</mi></msub><msub><mi>F</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>+</mo></msub><msub><mo>⊗</mo> <mi>X</mi></msub><msub><mi>F</mi> <mo>−</mo></msub><mo stretchy="false">)</mo><msub><mo>⊕</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mo>−</mo></msub><msub><mo>⊗</mo> <mi>X</mi></msub><msub><mi>F</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [E_+, E_-] \cdot [F_+, F_-] \;\coloneqq\; [ (E_+ \otimes_X F_+) \oplus_X (E_- \otimes_X F_-) \,,\, (E_+ \otimes_X F_-) \oplus_X (E_- \otimes_X F_+) ] \,. </annotation></semantics></math></div> <p>Accordingly the ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(K(X), +,\cdot)</annotation></semantics></math> is also called the <em>K-theory ring</em> 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> <div class="num_remark" id="FunctorialityOfKGroup"> <h6 id="remark_3">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/functor">functoriality</a> of the K-theory ring assignment)</strong></p> <p>Let <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 \longrightarrow Y</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. The operation of <a class="existingWikiWord" href="/nlab/show/pullback+of+vector+bundles">pullback of vector bundles</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Vect</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f^\ast \;\colon\; Vect(Y) \longrightarrow Vect(X) </annotation></semantics></math></div> <p>is compatible with <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> as well as with <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product of vector bundles</a> and hence descends to a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f^\ast \;\colon\; K(Y) \longrightarrow K(X) </annotation></semantics></math></div> <p>between the K-theory rings from remark <a class="maruku-ref" href="#KTheoryRing"></a>. Moreover, for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi><mover><mo>⟶</mo><mi>g</mi></mover><mi>Z</mi></mrow><annotation encoding="application/x-tex"> X \overset{f}{\longrightarrow} Y \overset{g}{\longrightarrow} Z </annotation></semantics></math></div> <p>two consecutive <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a>, then the consecutive pullback of the vector bundle is isomorphic to the pullback along the composite map, which means that on K-group pullback preserves composition</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>g</mi><mo>∘</mo><mi>f</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><msup><mi>f</mi> <mo>*</mo></msup><mo>∘</mo><msup><mi>g</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (g \circ f)^\ast = f^\ast \circ g^\ast \;\colon\; K(Z) \longrightarrow K(X) \,. </annotation></semantics></math></div> <p>Finally, of course pullback along an <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">id_X \colon X \to X</annotation></semantics></math> is the identity group homomorphism.</p> <p>In summary this says that the assignment of K-groups to topological spaces is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Top</mi> <mi>op</mi></msup><mo>⟶</mo><mi>CRing</mi></mrow><annotation encoding="application/x-tex"> K(-) \;\colon\; Top^{op} \longrightarrow CRing </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> to the category <a class="existingWikiWord" href="/nlab/show/CRing">CRing</a> of <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a>.</p> </div> <p>We consider next the image of plain vector bundles in <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundles">virtual vector bundles</a>:</p> <div class="num_prop" id="StableEquivalenceOfVectorBundles"> <h6 id="definition_4">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/stable+equivalence+of+vector+bundles">stable equivalence of vector bundles</a>)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. Define an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∼</mo> <mi>stable</mi></msub></mrow><annotation encoding="application/x-tex">\sim_{stable}</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by declaring two vector bundles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><msub><mi>E</mi> <mn>2</mn></msub><mo>∈</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_1 E_2 \in Vect(X)</annotation></semantics></math> to be equivalent if there exists a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X \times k^n</annotation></semantics></math> of some <a class="existingWikiWord" href="/nlab/show/rank">rank</a> <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> such that after <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> with this <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>, both bundles become <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo>∼</mo> <mi>stable</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><munder><mo>∃</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mrow><mo>(</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo>⊕</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>E</mi> <mn>2</mn></msub><msub><mo>⊕</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( E_1 \sim_{stable} E_2 \right) \;\Leftrightarrow\; \underset{n \in \mathbb{N}}{\exists} \left( E_1 \oplus_X (X \times k^n) \;\simeq\; E_2 \oplus_X (X \times k^n) \right) \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><msub><mo>∼</mo> <mi>stable</mi></msub><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_1 \sim_{stable} E_2</annotation></semantics></math> we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math> are <em>stably equivalent vector bundles</em>.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/image">image</a> of plain <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> in <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundles">virtual vector bundles</a>)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. There is a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/semigroups">semigroups</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><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mn>0</mn></msup><mo stretchy="false">]</mo><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Vect(X)_{/\sim} & \longrightarrow & K(X) \\ [E_1] &\overset{\phantom{AAA}}{\mapsto}& \left( [E_1], [X \times k^0] \right) } </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a> (def. <a class="maruku-ref" href="#SemigroupOfIsomorphismClassesOfTopologicalVectorBundlesOnX"></a>) to the K-group 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> (def. <a class="maruku-ref" href="#KGroupByGrothendieckGroup"></a> ).</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>, then the <a class="existingWikiWord" href="/nlab/show/image">image</a> of this function is the <a class="existingWikiWord" href="/nlab/show/stable+equivalence+classes+of+vector+bundles">stable equivalence classes of vector bundles</a> (def. <a class="maruku-ref" href="#StableEquivalenceOfVectorBundles"></a>), hence this function factors as an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> onto <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><msub><mo>∼</mo> <mi>stable</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">Vect(X)_{/\sim_{stable}}</annotation></semantics></math> followed by an <a class="existingWikiWord" href="/nlab/show/injection">injection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>⟶</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><msub><mo>∼</mo> <mi>stable</mi></msub></mrow></msub><mo>↪</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Vect(X)_{/\sim} \longrightarrow Vect(X)_{/\sim_{stable}} \hookrightarrow K(X) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The homomorphism of <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>→</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Vect(X)_{/\sim} \to K(X)</annotation></semantics></math> is the one given by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/Grothendieck+group+of+a+commutative+monoid">Grothendieck group</a> construction (<a href="Grothendieck+group+of+a+commutative+monoid#UniversalProperty">this prop.</a>).</p> <p>By definition of the <a class="existingWikiWord" href="/nlab/show/Grothendieck+group+of+a+commutative+monoid">Grothendieck group</a> (<a href="Grothendieck+group+of+a+commutative+monoid#GrothendieckGroupViaQuotientOfCartesianProduct">this def.</a>), two elements of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mn>0</mn></msup><mo stretchy="false">]</mo><mo>)</mo></mrow><mphantom><mi>AA</mi></mphantom><mtext>and</mtext><mphantom><mi>AA</mi></mphantom><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mn>0</mn></msup><mo stretchy="false">]</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( [E_1], [X \times k^0] \right) \phantom{AA} \text{and} \phantom{AA} \left( [E_2], [X \times k^0] \right) </annotation></semantics></math></div> <p>are equivalent precisely if there exist vector bundles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F_2</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo>⊕</mo> <mi>X</mi></msub><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>2</mn></msub><msub><mo>⊕</mo> <mi>X</mi></msub><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( [ E_1 \oplus_X F_1 ], [ F_1] \right) \;=\; \left( [E_2 \oplus_X F_2], [F_2] \right) \,. </annotation></semantics></math></div> <p>First of all this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub><mo>≃</mo><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F_1 \simeq F_2</annotation></semantics></math>, hence is equivalent to the existence of a vector bundle <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> such that</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>E</mi> <mn>1</mn></msub><mo>⊕</mo><mi>F</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo>⊕</mo><mi>F</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [E_1 \oplus F] \;=\; [E_2 \oplus F] \,. </annotation></semantics></math></div> <p>Now, by the assumption 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> is compact Hausdorff, lemma <a class="maruku-ref" href="#DirectSumHasInverseUpToTrivialBundle"></a> implies that there exists a vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>F</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde F</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><msub><mo>⊕</mo> <mi>X</mi></msub><mover><mi>F</mi><mo stretchy="false">˜</mo></mover><mo>≃</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> F \oplus_X \tilde F \simeq X \times k^n </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a> of some <a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank</a> <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>. This means that the above is equivalent already to the existence of an <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> such that</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>E</mi> <mn>1</mn></msub><mo>⊕</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo>⊕</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [E_1 \oplus (X \times k^n)] \;=\; [E_2 \oplus (X \times k^n)] \,. </annotation></semantics></math></div> <p>This is the definition of stable equivalence from def. <a class="maruku-ref" href="#StableEquivalenceOfVectorBundles"></a>.</p> </div> <h3 id="ReducedKGroup">The reduced K-group</h3> <div class="num_defn" id="KernelReducedKGroup"> <h6 id="definition_5">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a>)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a>, hence a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> equipped with a choice of point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>, hence with a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mi>x</mi></msub><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">const_x \colon \ast \to X</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/point+space">point space</a>.</p> <p>By the functoriality of the K-groups (remark <a class="maruku-ref" href="#FunctorialityOfKGroup"></a>) this induces a group homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>const</mi> <mi>x</mi> <mo>*</mo></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>K</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> const_x^\ast \;\colon\; K(X) \longrightarrow K(\ast) </annotation></semantics></math></div> <p>given by restricting a virtual vector bundle to the basepoint.</p> <p>The <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of this map is called the <em><a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> group</em> 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>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math>, denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>ker</mi><mo stretchy="false">(</mo><msubsup><mi>const</mi> <mi>x</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde{K}(X) \;\coloneqq\; ker(const_x^\ast) \,. </annotation></semantics></math></div></div> <div class="num_example" id="ExpressingPlainKTHeoryGroupInTermsOfReducedKTheoryGroup"> <h6 id="example_2">Example</h6> <p><strong>(expressing plain K-groups as reduced K-groups)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>*</mo></msub><mo>≔</mo><mi>X</mi><mo>⊔</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X_* \coloneqq X \sqcup \ast</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> with the <a class="existingWikiWord" href="/nlab/show/point+space">point space</a>, and regard this as a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a> with base point the adjoined point.</p> <p>Then the reduced K-theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">X_+</annotation></semantics></math> is the plain 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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde{K}(X_+) \simeq K(X) \,. </annotation></semantics></math></div> <p>Because every <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> on <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 \sqcup \ast</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> of one that has <a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank</a> zero on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> and one that has rank zero 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> (<a href="direct+sum+of+vector+bundles#DirectSumOnDisjointUnionSpace">this example.</a>)</p> </div> <div class="num_remark" id="RestrictionInKTheoryToPointComputesVirtualRank"> <h6 id="remark_4">Remark</h6> <p><strong>(restriction in K-theory to the point computes virtual rank)</strong></p> <p>By example <a class="maruku-ref" href="#KGroupOfThePoint"></a> we have that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">K(\ast) \simeq \mathbb{Z}</annotation></semantics></math>;</p> </li> <li> <p>under this identification the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>const</mi> <mi>x</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">const_x^\ast</annotation></semantics></math> is the assignment of <em>virtual rank</em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>const</mi> <mi>x</mi> <mo>*</mo></msubsup></mrow></mover></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd><mi>ℤ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>F</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mo stretchy="false">[</mo><msub><mi>E</mi> <mi>x</mi></msub><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><msub><mi>F</mi> <mi>x</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ K(X) &\overset{const_x^\ast}{\longrightarrow}& K(\ast) &\overset{\simeq}{\to}& \mathbb{Z} \\ [E]- [F] &\overset{\phantom{AAA}}{\mapsto}& [E_x] - [F_x] &\mapsto& rk_x(E) - rk_x(F) } </annotation></semantics></math></div></li> </ol> </div> <div class="num_defn" id="VanishingAtInfinity"> <h6 id="remark_5">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanishing at infinity</a>)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, then a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</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><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \longrightarrow \mathbb{R} </annotation></semantics></math></div> <p>is said to <a class="existingWikiWord" href="/nlab/show/vanishing+at+infinity">vanish at infinity</a> if it <a class="existingWikiWord" href="/nlab/show/extension">extends</a> by zero to 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><msup><mi>X</mi> <mo>*</mo></msup><mo>≔</mo><mo stretchy="false">(</mo><mi>X</mi><mo>⊔</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo><mo>,</mo><msub><mi>τ</mi> <mi>cpt</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X^* \coloneqq (X \sqcup \{\infty\}, \tau_{cpt})</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>ℝ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mi>x</mi><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>f</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>X</mi></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>x</mi><mo>=</mo><mn>∞</mn></mtd></mtr></mtable></mrow></mrow></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><msup><mi>X</mi> <mo>*</mo></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{f}{\longrightarrow}& \mathbb{R} \\ \downarrow & \nearrow_{\mathrlap{ x \mapsto \left\{ \array{ f(x) &\vert& x \in X \\ 0 &\vert& x = \infty } \right. }} \\ X^\ast } </annotation></semantics></math></div> <p>Now 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><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">X^\ast</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> (by <a href="one-point+compactification#OnePointExtensionIsCompact">this prop.</a> and <a href="one-point+compactification#HausdorffOnePointCompactification">this prop.</a>) and canonically a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a> with basepoint the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mo>∈</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\infty \in X^\ast</annotation></semantics></math>.</p> <p>Moreover, every <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> arises this way as the <a class="existingWikiWord" href="/nlab/show/one-point+compactification">one-point compactification</a> of the <a class="existingWikiWord" href="/nlab/show/complement">complement</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of any of its points: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∖</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">X \simeq (X \setminus \{x\})^\ast</annotation></semantics></math> (by <a href="one-point+compactification#CompactHausdorffSpaceIsCompactificationOfComplementOfAnyPoint">this remark</a>).</p> <p>Since <a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a>, this complement <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∖</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \setminus \{x\} \subset X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, and every locally compact Hausdorff spaces arises this way (by <a href="one-point+compactification#InclusionIntoOnePointExtensionIsOpenEmbedding">this prop.</a>).</p> <p>Therefore one may think of the <a class="existingWikiWord" href="/nlab/show/reduced+K-groups">reduced K-groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math> (def. <a class="maruku-ref" href="#KernelReducedKGroup"></a>) of compact Hausdorff spaces as the those K-groups of locally compact Hausdorff spaces which “vanish at infinity”.</p> </div> <div class="num_remark" id="FunctorialityOfReducedKGroups"> <h6 id="remark_6">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/functor">functoriality</a> of the reduced K-groups)</strong></p> <p>By the functoriality of the unreduced K-groups (remark <a class="maruku-ref" href="#FunctorialityOfKGroup"></a>) on (the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite</a> of) the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of all topological spaces, the reduced K-groups (def. <a class="maruku-ref" href="#KernelReducedKGroup"></a>) becomes functorial on the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Top^{\ast/}</annotation></semantics></math> of <em><a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a></em> (whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are the <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> that preserve the base-point):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>⟶</mo><mi>Ab</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde{K} \;\colon\; (Top^{\ast/})^{op} \longrightarrow Ab \,. </annotation></semantics></math></div> <p>This follows by the functoriality of the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> construction (which in turn follows by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the kernel):</p> <p>For <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>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,y)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> and <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 \longrightarrow Y</annotation></semantics></math> a continuous function which preserves basepoints <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f(x) = y</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ker</mi><mo stretchy="false">(</mo><msubsup><mi>const</mi> <mi>x</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>const</mi> <mi>x</mi> <mo>*</mo></msubsup></mrow></mover></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mo>∃</mo><mo>!</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ker</mi><mo stretchy="false">(</mo><msubsup><mi>const</mi> <mi>y</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>const</mi> <mi>x</mi> <mo>*</mo></msubsup></mrow></mover></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ ker(const_x^\ast) &\longrightarrow& K(X) &\overset{const_x^\ast}{\longrightarrow}& K(\{x\}) \\ \uparrow^{\mathrlap{\exists !}} && \uparrow^{\mathrlap{f^\ast}} && \uparrow^{\mathrlap{f^\ast}} \\ ker(const_y^\ast) &\longrightarrow& K(Y) &\overset{const_x^\ast}{\longrightarrow}& K(\{y\}) } \,. </annotation></semantics></math></div></div> <div class="num_prop" id="KGrupDirectSummandReducedKGroup"> <h6 id="proposition_3">Proposition</h6> <p><strong>(over <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct summand</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math>)</strong></p> <p>If <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>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> then the defining <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> groups (def. <a class="maruku-ref" href="#KernelReducedKGroup"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>↪</mo><mphantom><mi>AAA</mi></mphantom></mover><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msubsup><mi>const</mi> <mi>x</mi> <mo>*</mo></msubsup></mrow></mover><mi>K</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to \tilde{K}(X) \overset{\phantom{AAA}}{\hookrightarrow} K(X) \overset{const_x^\ast}{\longrightarrow} K(\ast) \simeq \mathbb{Z} \to 0 </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/split+exact+sequence">splits</a> and thus yields an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, which is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mo>≃</mo><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>ℤ</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo><mo>,</mo><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>−</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ K(X) &\overset{\phantom{A}\simeq \phantom{A}}{\longrightarrow}& \tilde{K}(X) \oplus \mathbb{Z} \\ [E] - [X \times k^n] &\overset{\phantom{AAA}}{\mapsto}& ([E], rk_x(E) - n) } \,. </annotation></semantics></math></div> <p>Here on the left we are using prop. <a class="maruku-ref" href="#OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle"></a> to represent any element of the K-group as a <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual</a> difference of a vector bundle <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> by a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">rk_x(E) \in \mathbb{N}</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/rank+of+a+vector+bundle">rank</a> of this vector bundle over the <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>.</p> <p>Equivalently this means that every element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math> decomposes as follows into a piece that has vanishing virtual rank over the connected component 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> and a virtual <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mi>n</mi></msup><mo stretchy="false">]</mo><mo>=</mo><munder><munder><mrow><mo>(</mo><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">]</mo><mo>)</mo></mrow><mo>⏟</mo></munder><mrow><mo>∈</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></munder><mo>−</mo><munder><munder><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><mi>n</mi><mo>−</mo><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">]</mo></mrow><mo>⏟</mo></munder><mrow><mo>∈</mo><mi>ℤ</mi><mo>⊂</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></munder><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [E]- [X \times k^n] = \underset{\in \tilde{K}(X) \subset K(X)}{\underbrace{\left( [E] - [X \times k^{rk_x(E)}] \right)}} - \underset{\in \mathbb{Z} \subset K(X) }{\underbrace{[X \times k^{n-rk_x(E)}]}} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By remark <a class="maruku-ref" href="#RestrictionInKTheoryToPointComputesVirtualRank"></a> the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>const</mi> <mi>x</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">const_x^\ast</annotation></semantics></math> is identified with the <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundles">virtual vector bundles</a> of vanishing virtual rank. By prop. <a class="maruku-ref" href="#OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle"></a> this kernel is identified with the elements of the form</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>E</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [E] - [X \times k^{rk_x(E)}] \,. </annotation></semantics></math></div></div> <div class="num_example" id="BottElement"> <h6 id="example_3">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Bott+element">Bott element</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a>, write</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><msub><mi>Vect</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> h \in Vect_{\mathbb{C}}(S^2) \longrightarrow K_{\mathbb{C}}(S^2) </annotation></semantics></math></div> <p>for the complex topological K-theory class of the <a class="existingWikiWord" href="/nlab/show/basic+complex+line+bundle+on+the+2-sphere">basic complex line bundle on the 2-sphere</a>. By prop. <a class="maruku-ref" href="#KGrupDirectSummandReducedKGroup"></a> its image in <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> is the <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>β</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>h</mi><mo>−</mo><mn>1</mn><mo>∈</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \beta \;\coloneqq \; h-1 \in \tilde K_{\mathbb{C}}(S^2) \,. </annotation></semantics></math></div> <p>This is known as the <em><a class="existingWikiWord" href="/nlab/show/Bott+element">Bott element</a></em>, due to its key role in the <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a> of complex topological K-theory, discussed <a href="#BottPeriodicities">below</a>.</p> </div> <p>In order to describe <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math> itself as an equivalence class, we consider the followign refinement of <a class="existingWikiWord" href="/nlab/show/stable+equivalence+of+vector+bundles">stable equivalence of vector bundles</a> (def. <a class="maruku-ref" href="#StableEquivalenceOfVectorBundles"></a>):</p> <div class="num_defn" id="EquivalenceRelationForReducedKTheory"> <h6 id="definition_6">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> for <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> on <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, define an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> on the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by declaring that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mo>∼</mo><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_1 \sim E_2</annotation></semantics></math> if there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>k</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>2</mn></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k_1, k_2 \in \mathbb{N}</annotation></semantics></math> such that there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a> between the <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">E_1</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a> <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> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">X \times \mathbb{R}^{k_1}</annotation></semantics></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math> with <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> <mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">X \times \mathbb{R}^{k_2}</annotation></semantics></math></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>E</mi> <mn>1</mn></msub><msub><mo>∼</mo> <mi>red</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⇔</mo><mrow><mo>(</mo><munder><mo>∃</mo><mrow><msub><mi>k</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>k</mi> <mn>2</mn></msub><mo>∈</mo><mi>ℕ</mi></mrow></munder><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo>⊕</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>2</mn></msub><msub><mo>⊕</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (E_1 \sim_{red} E_2) \Leftrightarrow \left( \underset{k_1,k_2 \in \mathbb{N}}{\exists} \left( (E_1 \oplus_X (X \times \mathbb{R}^{k_1}) \;\simeq\; (E_2 \oplus_X (X \times \mathbb{R}^{k_2}) \right) \right) \,. </annotation></semantics></math></div> <p>The operation of <a class="existingWikiWord" href="/nlab/show/direct+sum+of+vector+bundles">direct sum of vector bundles</a> descends to these quotients</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>E</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>1</mn></msub><msub><mo>⊕</mo> <mi>X</mi></msub><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [E_1] + [E_2] \;\coloneqq\; [ E_1 \oplus_X E_2 ] </annotation></semantics></math></div> <p>to yield a commutative <a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><msub><mo>∼</mo> <mi>red</mi></msub></mrow></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left(Vect(X)_{/\sim_{red}}, +\right) \,. </annotation></semantics></math></div></div> <div class="num_prop" id="ReducedKEquivalenceRelationVerified"> <h6 id="proposition_4">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+Hausdorff+space">compact Hausdorff space</a> then the commutative <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><msub><mo>∼</mo> <mi>red</mi></msub></mrow></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Vect(X)_{/\sim_{red}}, +)</annotation></semantics></math> from def. <a class="maruku-ref" href="#EquivalenceRelationForReducedKTheory"></a> is already an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> and is in fact <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math> (def. <a class="maruku-ref" href="#KernelReducedKGroup"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><msub><mo>∼</mo> <mi>red</mi></msub></mrow></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde{K}(X) \simeq (Vect(X)_{/\sim_{red}}, +) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By prop. <a class="maruku-ref" href="#KGrupDirectSummandReducedKGroup"></a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math> is the subgroup of the <a class="existingWikiWord" href="/nlab/show/Grothendieck+group+of+a+commutative+monoid">Grothendieck group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math> on the elements of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo><mo>−</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[E]- [X \times k^{rk_x(E)}]</annotation></semantics></math>, which are clearly entirely determined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>E</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding="application/x-tex">[E] \in Vect(X)_{/\sim}</annotation></semantics></math>. Hence we need to check if the equivalence relation of the Gorthendieck goup coincides with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∼</mo> <mi>red</mi></msub></mrow><annotation encoding="application/x-tex">\sim_{red}</annotation></semantics></math> on these representatives.</p> <p>The relation in the Grothendieck group is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>∼</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mo>⇔</mo><mrow><mo>(</mo><munder><mo>∃</mo><mrow><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow></munder><mrow><mo>(</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>H</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( [E_1] \sim [E_2] \right) \Leftrightarrow \left( \underset{[G], [H] \in Vect(X)_{/\sim}}{\exists} \left( ( [E_1]+ [G], [X \times k^{rk_x(E_1)}] + [G] ) \;=\; ( [E_2] + [H], [X \times k^{rk_x(E_2)}] + [H] ) \right) \right) </annotation></semantics></math></div> <p>As before, in prop. <a class="maruku-ref" href="#OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle"></a> we may assume without restriction that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">G = X \times k^{n_1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">H = X \times k^{n_2}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundles">trivial vector bundles</a>. Then the above equality on the first component</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>E</mi> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>+</mo><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [E_1] + [X \times k^{n_1}] = [E_2] + [X \times k^{n_2}] </annotation></semantics></math></div> <p>is the one that defines <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∼</mo> <mi>red</mi></msub></mrow><annotation encoding="application/x-tex">\sim_{red}</annotation></semantics></math>, and since isomorphic vector bundles necessarily have the same rank, it implies the equality of the second component.</p> </div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/non-unital+ring">non-unital</a> <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math>)</strong></p> <p>In view of the <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> structure on the K-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math> from remark <a class="maruku-ref" href="#KTheoryRing"></a>, the reduced K-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math> from def. <a class="maruku-ref" href="#KernelReducedKGroup"></a>, being the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of a ring <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> (remark <a class="maruku-ref" href="#FunctorialityOfKGroup"></a>) is an ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math>, hence itself a <a class="existingWikiWord" href="/nlab/show/non-unital+ring">non-unital</a> <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>.</p> <p>(The ring unit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math> is the class <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><msup><mi>k</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo>×</mo><msup><mi>k</mi> <mn>0</mn></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X \times k^1, X \times k^0]</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial</a> <a class="existingWikiWord" href="/nlab/show/line+bundle">line 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>, which has virtual rank 1, and hence is not in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math>.)</p> </div> <h3 id="TheRelativeKGroup">The relative K-group</h3> <div class="num_defn" id="RelativeKTheory"> <h6 id="definition_7">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/relative+K-theory">relative K-theory</a>)</strong></p> <p>Let</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a>.</p> </li> </ol> <p>Then the <em><a class="existingWikiWord" href="/nlab/show/relative+K-theory">relative K-theory</a> group of the pair <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>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,A)</annotation></semantics></math></em>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X,A)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> group (def. <a class="maruku-ref" href="#KernelReducedKGroup"></a>) of the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X/A</annotation></semantics></math> (<a href="quotient+space#QuotientBySubspace">this def.</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K(X,A) \;\coloneqq\; \tilde K(X/A) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="RelatveKTheoryReducesToBareKTheoryAndToReducedKTheory"> <h6 id="example_4">Example</h6> <p><strong>(expressing plain and reduced K-theory in terms of relative K-theory)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/relative+K-theory">relative K-theory</a> construction from def. <a class="maruku-ref" href="#RelativeKTheory"></a> reduces in special cases to the plain K-theory group and to the <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> group.</p> <p>Recall that for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>∅</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A = \emptyset \subset X</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>∅</mi><mo>=</mo><msub><mi>X</mi> <mo>+</mo></msub><mo>=</mo><mi>X</mi><mo>⊔</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X/\emptyset = X_+ = X \sqcup \ast</annotation></semantics></math> (by <a href="quotient+space#QuotientBySubspace">this example</a>). Therefore:</p> <ol> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>∅</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A = \emptyset \subset X</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>=</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>⊔</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X,\emptyset) = \tilde K(X \sqcup \ast) \simeq K(X)</annotation></semantics></math> (example <a class="maruku-ref" href="#ExpressingPlainKTHeoryGroupInTermsOfReducedKTheoryGroup"></a>);</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A = \{x\} \subset X</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X, \{x\}) = \tilde K(X/\{x\}) = \tilde{K}(X)</annotation></semantics></math>.</p> </li> </ol> </div> <h3 id="TheGradedKGroups">The graded K-groups</h3> <p>The (reduced) K-theory groups of reduced suspensions of pointed space are called the “K-group in degree 1”:</p> <div class="num_defn" id="GradedKGroups"> <h6 id="definition_8">Definition</h6> <p><strong>(graded K-groups)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a> write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde K^1(X) \;\coloneqq\; \tilde K(\Sigma X) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> of the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</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> <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+Hausdorff+space">compact Hausdorff space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K^1(X,A) \coloneqq \tilde K( \Sigma(X/A) ) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> of the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a> of the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>.</p> <p>We say these are the K(-cohomology)-groups in degree 1. For emphasis one says that the original K-groups are in degree zero and writes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mphantom><mi>AAAA</mi></mphantom><msup><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mphantom><mi>AAA</mi></mphantom><msup><mi>K</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K^0(X) \coloneqq K(X) \phantom{AAAA} \tilde K^0(X) \coloneqq \tilde{K}(X) \phantom{AAA} K^0(X,A) \coloneqq K(X,A) \,. </annotation></semantics></math></div> <p>The groups are collected to the <em>graded K-groups</em>, which are the <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde K^\bullet(X) \coloneqq \tilde K^0(X) \oplus \tilde K^1(X) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>K</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊕</mo><msup><mi>K</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>.</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K^\bullet(X,A) \coloneqq K^0(X,A) \oplus K^1(X.A) </annotation></semantics></math></div> <p>regarded as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-graded groups.</p> <p>Under <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product of vector bundles</a> this becomes a non-unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>- <a class="existingWikiWord" href="/nlab/show/graded-commutative+ring">graded-commutative ring</a> (discussed <a href="#GradedRingStructure">below</a>).</p> </div> <p>Recall from example <a class="maruku-ref" href="#ExpressingPlainKTHeoryGroupInTermsOfReducedKTheoryGroup"></a> and from example <a class="maruku-ref" href="#RelatveKTheoryReducesToBareKTheoryAndToReducedKTheory"></a> the identifications of plain, reduced and relative K-groups, which with the degree-zero notation from def. <a class="maruku-ref" href="#GradedKGroups"></a> read:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>+</mo></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>K</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K^0(X, \emptyset) \simeq \tilde K^0(X_+) \simeq K^0(X) </annotation></semantics></math></div> <p>The analogue is true for the K-groups in degree 1 from def. <a class="maruku-ref" href="#GradedKGroups"></a>, though this is no longer completely trivial:</p> <h3 id="ForNonCompactSpaces">For non-compact spaces</h3> <p>By prop. <a class="maruku-ref" href="#OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle"></a> the topological K-theory groups of <a class="existingWikiWord" href="/nlab/show/compact+topological+spaces">compact topological spaces</a> are <a class="existingWikiWord" href="/nlab/show/representable+functor">represended</a> by <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> into the <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>O</mi><mo>×</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">B O \times \mathbb{Z}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo>×</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">B U \times \mathbb{Z}</annotation></semantics></math>, respectively (def. <a class="maruku-ref" href="#BUn"></a>).</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>X</mi><mtext>compact</mtext><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>→</mo><mi>B</mi><mi>U</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">]</mo><mphantom><mi>AAAAA</mi></mphantom><msub><mi>K</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>O</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( X \text{compact} \right) \;\Rightarrow\; \left( K_{\mathbb{C}}(X) \simeq [X \to B U \times \mathbb{Z}] \phantom{AAAAA} K_{\mathbb{R}}(X) \simeq [X, B O \times \mathbb{Z}] \right) \,. </annotation></semantics></math></div> <p>There are various ways of generalizing this situation to non-compact spaces:</p> <div class="num_defn" id="GrothendieckGroupKTheory"> <h6 id="definition_9">Definition</h6> <p><strong>(Grothendieck group topological K-theory)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a> construction on the <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mo>,</mo><mo>⊕</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Vect(X)/_\sim, \oplus)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a> makes sense for every <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. For non-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> this is usually just called the “Grothendieck group of vector bundles 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>”, sometimes denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>GrothGroup</mi><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mo>,</mo><mo>⊕</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{K}(X) \coloneqq GrothGroup( Vect(X)/_\sim, \oplus ) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="RepresntableTopologicalKTheory"> <h6 id="definition_10">Definition</h6> <p><strong>(representable topological K-theory)</strong></p> <p>The group of <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> into a (classifying) space is of course well defined for any domain space, hence for any topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> one may set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>rep</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>U</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">]</mo><mphantom><mi>AAAAA</mi></mphantom><msub><mi>K</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>rep</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>O</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> K_{\mathbb{C}}(X)_{rep} \;\coloneqq\; [X,B U \times \mathbb{Z}] \phantom{AAAAA} K_{\mathbb{R}}(X)_{rep} \;\coloneqq\; [X,B O \times \mathbb{Z}] </annotation></semantics></math></div> <p>This is called <em>representable K-theory</em>.</p> </div> <p>Representable K-theory over <a class="existingWikiWord" href="/nlab/show/paracompact+topological+spaces">paracompact topological spaces</a> was discussed in (<a href="#Karoubi70">Karoubi 70</a>).</p> <div class="num_defn" id="InverseLimitTopologicalKTheory"> <h6 id="definition_11">Definition</h6> <p><strong>(inverse limit topological K-theory)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> with the structure of a <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>, hence a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> (“<a class="existingWikiWord" href="/nlab/show/direct+limit">direct limit</a>”) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>n</mi></msub><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X \simeq \underset{\longrightarrow}{\lim}_n X_n</annotation></semantics></math> such that each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_n</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finite+cell+complex">finite cell complex</a>, hence in particular a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a>. Then the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> (<a class="existingWikiWord" href="/nlab/show/inverse+limit">inverse limit</a>) of the corresponding K-group</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>invl</mi></msub><mo>≔</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>n</mi></msub><mi>K</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(X)_{invl} \coloneqq \underset{\longleftarrow}{\lim}_n K(X_n) </annotation></semantics></math></div> <p>is called the <em>inverse limit K-theory</em> 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> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>No two of these definitions are equivalent to each other on all of their domain of defintion (e.g. <a href="#AndersonHodgkin68">Anderson-Hodgkin 68</a>, <a href="#JackowskiOliver">Jackowski-Oliver</a>).</p> <p>Representable and direct limit K-theory of spaces that are <a class="existingWikiWord" href="/nlab/show/sequential+colimits">sequential colimits</a> of <a class="existingWikiWord" href="/nlab/show/compact+spaces">compact spaces</a> differ in general by a <a class="existingWikiWord" href="/nlab/show/lim%5E1">lim^1</a>-term (<a href="#SegalAtiyah69">Segal-Atiyah 69, prop. 4.1</a>).</p> <h2 id="Examples">Examples</h2> <div class="num_example"> <h6 id="example_5">Example</h6> <p><strong>(topological K-theory ring of the <a class="existingWikiWord" href="/nlab/show/point+space">point space</a>)</strong></p> <p>We have already seen in example <a class="maruku-ref" href="#KGroupOfThePoint"></a> that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K(\ast) \simeq \mathbb{Z} \,. </annotation></semantics></math></div></div> <div class="num_example" id="ComplexTopologicalKTheoryOfTheCircle"> <h6 id="example_6">Example</h6> <p><strong>(complex topological K-theory ring of the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>)</strong></p> <p>Since the complex <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n,\mathbb{C})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/path-connected+topological+space">path-connected</a> (<a href="general+linear+group#ConnectednessOfGeneralLinearGroup">this prop.</a>) and hence the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B GL(n,\mathbb{C})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply</a>-connected, hence its <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> is trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\pi_1(B GL(n,\mathbb{C})) \simeq [S^1, B GL(n,\mathbb{C})] = 1</annotation></semantics></math>. Accordingly, all <a class="existingWikiWord" href="/nlab/show/complex+vector+bundles">complex vector bundles</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a>.</p> <p>It follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mphantom><mi>AA</mi></mphantom><mtext>and</mtext><mphantom><mi>AA</mi></mphantom><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K_{\mathbb{C}}(S^1) \simeq \mathbb{Z} \phantom{AA} \text{and} \phantom{AA} \tilde K_{ \mathbb{C} }(S^1) \simeq 0 \,. </annotation></semantics></math></div></div> <div class="num_example" id="TopologicalKTheoryRingOfThe2Sphere"> <h6 id="example_7">Example</h6> <p><strong>(complex topological K-theory ring of the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a>)</strong></p> <p>For <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 = \ast</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/point+space">point space</a>, the <a class="existingWikiWord" href="/nlab/show/fundamental+product+theorem+in+topological+K-theory">fundamental product theorem in topological K-theory</a> <a class="maruku-ref" href="#FundamentalProductTheorem"></a> states that the homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>h</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>h</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>h</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z}[h]/((h-1)^2) &\longrightarrow& K_{\mathbb{C}}(S^2) \\ h &\mapsto& h } </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p>This means that the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(h-1)^2 = 0</annotation></semantics></math> satisfied by the <a class="existingWikiWord" href="/nlab/show/basic+complex+line+bundle+on+the+2-sphere">basic complex line bundle on the 2-sphere</a> (<a href="basic+complex+line+bundle+on+the+2-sphere#TensorRelationForBasicLineBundleOn2Sphere">this prop.</a>) is the <em>only</em> relation is satisfies in topological K-theory.</p> <p>Notice that the underlying <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>h</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[h]/((h-1)^2)</annotation></semantics></math> is two <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> copies of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo>⊕</mo><mi>ℤ</mi><mo>=</mo><mo stretchy="false">⟨</mo><mn>1</mn><mo>,</mo><mi>h</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> K_{\mathbb{C}}(S^2) \simeq \mathbb{Z} \oplus \mathbb{Z} = \langle 1, h\rangle </annotation></semantics></math></div> <p>one copy spanned by the <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial</a> <a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex line bundle</a> on the 2-sphere, the other spanned by the <a class="existingWikiWord" href="/nlab/show/basic+complex+line+bundle+on+the+2-sphere">basic complex line bundle on the 2-sphere</a>. (In contrast, the underlying abelian group of the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">[</mo><mi>h</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}[h]</annotation></semantics></math> has infinitely many copies of <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">\mathbb{Z}</annotation></semantics></math>, one for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>h</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">h^n</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>).</p> <p>It follows (by <a href="topological+K-theory#KGrupDirectSummandReducedKGroup">this prop.</a>) that the <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> group of the 2-sphere is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde K_{\mathbb{C}}(S^2) \simeq \mathbb{Z} \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_8">Example</h6> <p><strong>(complex topological K-theory of the <a class="existingWikiWord" href="/nlab/show/torus">torus</a>)</strong></p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/torus">torus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1 \times S^1</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> of the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> with itself (with the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>).</p> <p>By example <a class="maruku-ref" href="#ReducedKTheoryOfProductSpace"></a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><munder><munder><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><mo>⏟</mo></munder><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow></munder><mo stretchy="false">)</mo><mo>⊕</mo><munder><munder><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde K(S^1 \times S^1) \simeq \tilde K(\underset{S^2}{\underbrace{S^1 \wedge S^1}}) \oplus \underset{= 0}{ \underbrace{ \tilde K(S^1) \oplus \tilde K(S^1) }} \,. </annotation></semantics></math></div> <p>Since the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> of the circle with itself is the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a>, and since, the complex K-theory of the circle vanishes by example <a class="maruku-ref" href="#ComplexTopologicalKTheoryOfTheCircle"></a>, this shows that the topological K-theory of the torus coincides with that of the 2-sphere:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K(S^1 \times S^1) \simeq K(S^2) \,. </annotation></semantics></math></div></div> <h2 id="properties">Properties</h2> <h3 id="homotopy_invariance">Homotopy invariance</h3> <div class="num_prop" id="KGroupsHomotopyInvariance"> <h6 id="proposition_5">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopy+invariance">homotopy invariance</a> of K-groups)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces">paracompact Hausdorff spaces</a>, and let</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><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> which is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>. Then the pullback operation on (reduced) K-groups along <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> (remark <a class="maruku-ref" href="#FunctorialityOfKGroup"></a>, remark <a class="maruku-ref" href="#FunctorialityOfReducedKGroups"></a>) is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f^\ast \;\colon\; K(Y) \overset{\simeq}{\longrightarrow} K(X) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^\ast \;\colon\; \tilde K(Y) \overset{\simeq}{\longrightarrow} \tilde{K}(X) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>This is an immediate consequence of the fact that over paracompact Hausdorff spaces isomorphism classes of topological vector bundles are homotopy invariant (<a href="topological+vector+bundle#HomotopyInvarianceOfIsomorphismClassesOfVectorBundles">this example.</a>)</p> </div> <h3 id="ExactSequences">Exact sequences</h3> <p>We discuss the <a class="existingWikiWord" href="/nlab/show/long+exact+sequences+in+cohomology">long exact sequences in cohomology</a> for topological K-theory.</p> <p>What makes these work in prop. <a class="maruku-ref" href="#ExactSequenceInReducedTopologicalKTheory"></a> below, turns out to be the following property of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a>:</p> <div class="num_lemma" id="VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace"> <h6 id="lemma_2">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial</a> over <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a> of <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> is <a class="existingWikiWord" href="/nlab/show/pullback+bundle">pullback</a> of bundle on <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a>.</p> <p>If a <a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>p</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">E \overset{p}{\to} X</annotation></semantics></math> is such that its restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mo stretchy="false">|</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">E\vert_A</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/trivializable+vector+bundle">trivializable</a>, then <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> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/pullback+bundle">pullback bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>q</mi> <mo>*</mo></msup><mi>E</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">q^\ast E'</annotation></semantics></math> of a topological vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>′</mo><mo>→</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">E' \to X/A</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>..</p> </div> <p>The <strong>proof</strong> of lemma <a class="maruku-ref" href="#VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace"></a> is given at <em><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a></em> <a href="topological+vector+bundle+VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace">here</a>. What makes that proof work, in turn, is the <a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a>, via <a href="topological+vector+bundle#IsomorphismOfVectorBundlesOnClosedSubsetOfCompactHausdorffSpaceExtendsToOpenNeighbourhoods">this lemma</a>.</p> <div class="num_prop" id="ExactSequenceInReducedTopologicalKTheory"> <h6 id="proposition_6">Proposition</h6> <p><strong>(exact sequence in reduced topological K-theory)</strong></p> <p>Let</p> <ul> <li> <p><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> be a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a></p> </li> </ul> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X/A</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> (<a href="quotient+space#QuotientBySubspace">this def.</a>). Denote the <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> of <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> inclusion and of <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> co-projection by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>⟶</mo><mi>i</mi></mover><mi>X</mi><mover><mo>⟶</mo><mi>q</mi></mover><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> A \overset{i}{\longrightarrow} X \overset{q}{\longrightarrow} X/A \,, </annotation></semantics></math></div> <p>respectively. Then the induced sequence of <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> groups (def. <a class="maruku-ref" href="#KernelReducedKGroup"></a>, remark <a class="maruku-ref" href="#FunctorialityOfReducedKGroups"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>q</mi> <mo>*</mo></msup></mrow></mover><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde K_{\mathbb{C}}(X/A) \overset{q^\ast}{\longrightarrow} \tilde K_{\mathbb{C}}(X) \overset{i^\ast}{\longrightarrow} \tilde K_{\mathbb{C}}(A) </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact</a>, meaning that they induce an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msup><mi>q</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ker</mi><mo stretchy="false">(</mo><msup><mi>i</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> im(q^\ast) \simeq ker(i^\ast) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/image">image</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>g</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">g^\ast</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">i^\ast</annotation></semantics></math>.</p> <p>Similarly the sequence of unreduced and <a class="existingWikiWord" href="/nlab/show/relative+K-groups">relative K-groups</a> (def. <a class="maruku-ref" href="#RelativeKTheory"></a>) is exact:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>q</mi> <mo>*</mo></msup></mrow></mover><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde K_{\mathbb{C}}(X/A) \overset{q^\ast}{\longrightarrow} K_{\mathbb{C}}(X) \overset{i^\ast}{\longrightarrow} K_{\mathbb{C}}(A) </annotation></semantics></math></div></div> <p>(e.g. <a href="#Wirthmuller12">Wirthmüller, 12, p. 32 (34 of 67)</a>, <a href="#Hatcher">Hatcher, prop. 2.9</a>)</p> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>First observe that both statements are equivalent to each other: By <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> of the construction of <a class="existingWikiWord" href="/nlab/show/kernels">kernels</a> the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</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><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>q</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>q</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mi>id</mi></munder></mtd> <mtd><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde K_{\mathbb{C}}(X/A) &\overset{q^\ast}{\longrightarrow}& \tilde K_{\mathbb{C}}(X) &\overset{i^\ast}{\longrightarrow}& \tilde K_{\mathbb{C}}(A) \\ {}^{\mathllap{id}}\downarrow && \downarrow && \downarrow \\ \tilde K_{\mathbb{C}}(X/A) &\overset{q^\ast}{\longrightarrow}& K_{\mathbb{C}}(X) &\overset{i^\ast}{\longrightarrow}& K_{\mathbb{C}}(A) \\ && \downarrow && \downarrow \\ && K_{\mathbb{C}}(\ast) &\underset{id}{\longrightarrow}& K_{\mathbb{C}}(\ast) } \,. </annotation></semantics></math></div> <p>Here the top vertical morphisms, the <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> inclusions, are <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>, and hence the top horizonal row is exact precisely if the middle horizontal row is.</p> <p>Hence it is sufficient to consider the top row. First of all the composite function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>X</mi><mover><mo>→</mo><mi>q</mi></mover><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A \overset{i}{\to} X \overset{q}{\to} X/A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/constant+function">constant function</a>, constant on the basepoint, and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mo>∘</mo><msup><mi>q</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">i^\ast \circ q^\ast</annotation></semantics></math> is a constant function, constant on zero. This says that we have an inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>im</mi><mo stretchy="false">(</mo><msup><mi>q</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⊂</mo><mi>ker</mi><mo stretchy="false">(</mo><msup><mi>i</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> im(q^\ast) \subset ker(i^\ast) \,. </annotation></semantics></math></div> <p>Hence it only remains to see for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in \tilde{K}(X)</annotation></semantics></math> a class with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">i^\ast(x) = 0</annotation></semantics></math> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><msup><mi>q</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x = q^\ast(y)</annotation></semantics></math> comes from a class on the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X/A</annotation></semantics></math>. But by compactness, the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> is given by a <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>−</mo><mi>rk</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E - rk(E)</annotation></semantics></math> (prop. <a class="maruku-ref" href="#OnCompactHausdorffVirtualVectorBundlesAreFormalDifferentcesWithATrivialBundle"></a>, prop. <a class="maruku-ref" href="#KGrupDirectSummandReducedKGroup"></a>). and by prop. <a class="maruku-ref" href="#ReducedKEquivalenceRelationVerified"></a> the triviality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>E</mi><mo>−</mo><mi>rk</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i^\ast(E- rk(E))</annotation></semantics></math> means that there is <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>*</mo><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><msub><mo>⊕</mo> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>×</mo><msup><mi>ℂ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i \ast(E) \oplus_A (A \times \mathbb{C}^n)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/trivializable+vector+bundle">trivializable vector bundle</a>. Therefore lemma <a class="maruku-ref" href="#VectorBundleOnClosedSubsetOfCompactHausdorffSpaceIsPullbackOfBundeOnQuotientSpace"></a> gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msub><mo>⊕</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>ℂ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E \oplus_X (X \times \mathbb{C}^n)</annotation></semantics></math> is isomorphic to the <a class="existingWikiWord" href="/nlab/show/pullback+bundle">pullback bundle</a> of a vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">E'</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X/A</annotation></semantics></math>. This proves the claim.</p> </div> <div class="num_cor" id="LongExactSequenceInReducedTopologicalKTheory"> <h6 id="corollary">Corollary</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> in <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced topological K-theory</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> inclusion, there is a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> groups of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>⟶</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>Σ</mi><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \cdots \longrightarrow \tilde K_{\mathbb{C}}(\Sigma (X/A)) \longrightarrow \tilde K_{\mathbb{C}}(\Sigma X) \longrightarrow \tilde K_{\mathbb{C}}( \Sigma A ) \longrightarrow \tilde K_{\mathbb{C}}(X/A) \longrightarrow \tilde K_{\mathbb{C}}(X) \longrightarrow \tilde K_{\mathbb{C}}(A) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Sigma(-)</annotation></semantics></math> denotes <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a>.</p> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>The sequence is induced by functoriality (remark <a class="maruku-ref" href="#FunctorialityOfReducedKGroups"></a>) from the long <a class="existingWikiWord" href="/nlab/show/cofiber+sequence">cofiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>⟶</mo><mi>i</mi></mover><mi>X</mi><mo>⟶</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∪</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> A \overset{i}{\longrightarrow} X \longrightarrow X \cup Cone(A) \longrightarrow (X \cup Cone(A)) \cup Cone(X) \longrightarrow ((X \cup Cone(A)) \cup Cone(X)) \cup (X \cup Cone(A)) \longrightarrow \cdots </annotation></semantics></math></div> <p>obtained by consecutively forming <a class="existingWikiWord" href="/nlab/show/mapping+cones">mapping cones</a>. By the discussion at <em><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">topological cofiber sequence</a></em> this may be rearranged as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>j</mi></mover></mtd> <mtd><mi>Cone</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Cone</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mfrac linethickness="0"><mrow><mtext>homotopy</mtext></mrow><mrow><mtext>equivalence</mtext></mrow></mfrac></mpadded></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mfrac linethickness="0"><mrow><mtext>homotopy</mtext></mrow><mrow><mtext>equivalence</mtext></mrow></mfrac></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mtd> <mtd></mtd> <mtd><mi>S</mi><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>S</mi><mi>i</mi></mrow></mover></mtd> <mtd><mi>S</mi><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{i}{\longrightarrow}& X &\overset{j}{\longrightarrow}& Cone(i) &\longrightarrow& Cone(j) \\ && && \downarrow {\mathrlap{\text{homotopy} \atop \text{equivalence}}} && \downarrow^{\mathrlap{\text{homotopy} \atop \text{equivalence}}} \\ && && X/A && S A &\overset{S i}{\longrightarrow}& S X &\longrightarrow& \cdots } </annotation></semantics></math></div> <p>(<a href="topological+cofiber+sequence#HomotopyEquivalenceSuspensionWithMappingConeOfMappingCone">this prop.</a> and <a href="topological+cofiber+sequence">this</a>).</p> <p>The claim hence follows by the <a class="existingWikiWord" href="/nlab/show/homotopy+invariance">homotopy invariance</a> of the K-groups (prop. <a class="maruku-ref" href="#KGroupsHomotopyInvariance"></a>).</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>We discuss some useful consequences of the <a class="existingWikiWord" href="/nlab/show/long+exact+sequences+in+cohomology">long exact sequences in cohomology</a>.</p> <div class="num_prop" id="DirectSumOfKTheoryGroupsOverRetracts"> <h6 id="proposition_7">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> decomposition of K-theory groups over <a class="existingWikiWord" href="/nlab/show/retractions">retractions</a>)</strong></p> <p>Let <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> be a (<a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a>) <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \subset X</annotation></semantics></math> a (<a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a>) <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a>, such that the <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover></mrow><annotation encoding="application/x-tex">A \overset{i}{\to}</annotation></semantics></math> X as a <a class="existingWikiWord" href="/nlab/show/retraction">retraction</a>, i.e. a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">r \colon X \to A</annotation></semantics></math> such that the <a class="existingWikiWord" href="/nlab/show/composition">composite</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mover><mo>⟶</mo><mi>i</mi></mover><mi>X</mi><mover><mo>⟶</mo><mi>r</mi></mover><mi>A</mi></mrow><annotation encoding="application/x-tex"> id_A \;\colon\; A \overset{i}{\longrightarrow} X \overset{r}{\longrightarrow} A </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a>.</p> <p>Then there is a splitting of the K-theory group 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> as a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of the K-theory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/relative+K-theory">relative K-theory</a> of the <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X/A</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(X) \;\simeq\; K(A) \oplus K(X,A) </annotation></semantics></math></div> <p>and in the pointed case a splitting of the <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde{K}(X) \;\simeq\; \tilde K(A) \oplus K(X,A) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>The long exact sequence from cor <a class="maruku-ref" href="#LongExactSequenceInReducedTopologicalKTheory"></a> together with the retraction yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><munderover><mrow></mrow><munder><mo>⟶</mo><mrow><msup><mi>r</mi> <mo>*</mo></msup></mrow></munder><mover><mo>⟵</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover></munderover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟵</mo><mrow></mrow></mover><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟵</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>A</mi><mo stretchy="false">)</mo><munderover><mrow></mrow><munder><mo>⟶</mo><mrow></mrow></munder><mover><mo>⟵</mo><mrow></mrow></mover></munderover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde K(A) \underoverset {\underset{r^\ast}{\longrightarrow}} {\overset{i^\ast}{\longleftarrow}} {} \tilde{K}(X) \overset{}{\longleftarrow} K(X,A) \longleftarrow \tilde K(\Sigma A) \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longleftarrow}} {} \tilde K(\Sigma X) \,. </annotation></semantics></math></div> <p>The splitting makes the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">i^\ast</annotation></semantics></math> and its suspension be <a class="existingWikiWord" href="/nlab/show/surjections">surjections</a>, so that the long exact sequence decomposes into <a class="existingWikiWord" href="/nlab/show/short+exact+sequences">short exact sequences</a> which are <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>⟵</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><munderover><mrow></mrow><munder><mo>⟶</mo><mrow><msup><mi>r</mi> <mo>*</mo></msup></mrow></munder><mover><mo>⟵</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover></munderover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟵</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟵</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \longleftarrow \tilde K(A) \underoverset {\underset{r^\ast}{\longrightarrow}} {\overset{i^\ast}{\longleftarrow}} {} \tilde{K}(X) \longleftarrow K(X,A) \longleftarrow 0 \,. </annotation></semantics></math></div></div> <div class="num_example" id="FiniteWedgePropertyForReducedTopologicalKTheory"> <h6 id="example_9">Example</h6> <p><strong>(finite <a class="existingWikiWord" href="/nlab/show/wedge+axiom">wedge axiom</a>)</strong></p> <p>Let <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>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,y)</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a> with <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∨</mo><mi>Y</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>x</mi><mo>∼</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X \vee Y \;\coloneqq\; (X \sqcup Y)/(x \sim y) </annotation></semantics></math></div> <p>(i.e. the <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient</a> of their <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> by re-identifying the base points).</p> <p>Then there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde K(X \vee Y) \simeq \tilde{K}(X) \oplus \tilde K(Y) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>We have <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo><mo>↪</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> X = X \times \{y\} \hookrightarrow X \times Y </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo>×</mo><mi>Y</mi><mo>↪</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Y = \{x\} \times Y \hookrightarrow (X \times Y) / (X \times \{y\}) \,. </annotation></semantics></math></div> <p>Applying prop. <a class="maruku-ref" href="#DirectSumOfKTheoryGroupsOverRetracts"></a> to each of these consecutively yields an isomorphism that establishes the claim:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde K(X \times Y) \simeq \tilde{K}(X) \oplus \tilde K( (X \times Y)/(X \times \{y\}) ) \simeq \tilde{K}(X) \oplus \tilde K(Y) \oplus \tilde K(X \wedge Y) \,. </annotation></semantics></math></div> <p>This proves the claim.</p> <p>Alternatively, we may again argue directly from the long exact sequence:</p> <p>Consider the subspace inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊂</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \subset X \vee Y \,. </annotation></semantics></math></div> <p>This is a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a> because its <a class="existingWikiWord" href="/nlab/show/complement">complement</a> is <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>Y</mi><mo stretchy="false">)</mo><mo>∖</mo><mi>X</mi><mo>=</mo><mi>Y</mi><mo>∖</mo><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">(X \vee Y) \setminus X = Y \setminus \{y\}</annotation></semantics></math> which is open because all points in a <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a> (which is in particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/separation+axiom">separated</a>) are closed. Moreover, by definition of <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a> the corresponding <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a> is <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><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>X</mi><mo>≃</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X \vee Y) / X \simeq Y \,. </annotation></semantics></math></div> <p>Similary for the inclusion of <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>. Hence in particular these inclusions and quotients are <a class="existingWikiWord" href="/nlab/show/retractions">retractions</a> in that they factor the <a class="existingWikiWord" href="/nlab/show/identity+maps">identity maps</a> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo>⟶</mo><mi>X</mi><mphantom><mi>AA</mi></mphantom><mtext>and</mtext><mphantom><mi>AA</mi></mphantom><msub><mi>id</mi> <mi>Y</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>⟶</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo>⟶</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> id_X \;\colon\; X \longrightarrow X \vee Y \longrightarrow X \phantom{AA} \text{and} \phantom{AA} id_Y \;\colon\; Y \longrightarrow X \vee Y \longrightarrow Y \,. </annotation></semantics></math></div> <p>By functoriality (remark <a class="maruku-ref" href="#FunctorialityOfReducedKGroups"></a>) this implies that similarly</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mtext>and</mtext><mphantom><mi>AA</mi></mphantom><msub><mi>id</mi> <mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> id_{\tilde{K}(X)} \;\colon\; \tilde{K}(X) \longrightarrow \tilde K(X \vee Y) \longrightarrow \tilde{K}(X) \phantom{AA} \text{and} \phantom{AA} id_{\tilde K(Y)} \;\colon\; \tilde K(Y) \longrightarrow \tilde K(X \vee Y) \longrightarrow \tilde K(Y) \,. </annotation></semantics></math></div> <p>In particular these maps are <a class="existingWikiWord" href="/nlab/show/injective+function">injections</a> and <a class="existingWikiWord" href="/nlab/show/surjective+function">surjections</a>, respectively.</p> <p>Therefore by prop. <a class="maruku-ref" href="#ExactSequenceInReducedTopologicalKTheory"></a> there are <a class="existingWikiWord" href="/nlab/show/short+exact+sequences">short exact sequences</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to \tilde{K}(X) \longrightarrow \tilde K(X \vee Y) \longrightarrow \tilde K(Y) \to 0 </annotation></semantics></math></div> <p>which are <a class="existingWikiWord" href="/nlab/show/split+exact+sequences">split exact</a>. This implies the claim.</p> </div> <h3 id="ExternalProducts">External product</h3> <div class="num_defn" id="ExternalTensorProductInKTheory"> <h6 id="definition_12">Definition</h6> <p><strong>(external product in K-theory)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. Then the <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product</a> of <a class="existingWikiWord" href="/nlab/show/topological+vector+bundles">topological vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <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><mo>⊠</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \boxtimes \;\colon\; Vect(X) \times Vect(Y) \longrightarrow Vect(X \times Y) </annotation></semantics></math></div> <p>induces on K-groups an <em>external product</em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⊠</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>K</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \boxtimes \;\colon\; K(X) \oplus K(Y) \longrightarrow K(X \times Y) </annotation></semantics></math></div></div> <p>We want to see that this restricts to an operation on <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a>. To this end we need the following proposition:</p> <div class="num_prop" id="ReducedKTheoryOfProductSpace"> <h6 id="proposition_8">Proposition</h6> <p><strong>(reduced K-theory of product space)</strong></p> <p>Let <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><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x_0)</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>y</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,y_0)</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a> with <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 \times Y</annotation></semantics></math> their <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> and <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 \wedge Y</annotation></semantics></math> their <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a>. Then there is an isomorphism of <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde K(X \times Y) \;\simeq\; \tilde K(X \wedge Y) \oplus \tilde{K}(X) \oplus \tilde K(Y) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>Be definition, the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> is the <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a> of the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> by the <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∧</mo><mi>Y</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X \wedge Y \;=\; (X \times Y) / (X \vee Y) </annotation></semantics></math></div> <p>for the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo>∨</mo><mi>Y</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>Y</mi></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><msub><mi>y</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>y</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \vee Y &\overset{i}{\longrightarrow}& X \times Y \\ x &\mapsto& (x, y_0) \\ y &\mapsto& (x_0, y) } \,. </annotation></semantics></math></div> <p>This quotient is still a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a> because <a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a> and and it is still <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff topological space</a> because <a class="existingWikiWord" href="/nlab/show/compact+subspaces+in+Hausdorff+spaces+are+separated+by+neighbourhoods+from+points">compact subspaces in Hausdorff spaces are separated by neighbourhoods from points</a>, so that the point <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>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X \vee Y)/ (X \vee Y) \in (X \times Y)/(X \vee Y)</annotation></semantics></math> is separated by open neighbourhoods from points in <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>Y</mi><mo stretchy="false">)</mo><mo>∖</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X \times Y) \setminus (X \vee Y)</annotation></semantics></math>.</p> <p>Hence corollary <a class="maruku-ref" href="#LongExactSequenceInReducedTopologicalKTheory"></a> yields a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>Σ</mi><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>∨</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde K(\Sigma(X \times Y)) \overset{\Sigma i^\ast}{\longrightarrow} \tilde K( (\Sigma X) \vee (\Sigma Y)) \longrightarrow \tilde K( X \wedge Y ) \longrightarrow \tilde K( X \times Y ) \overset{i^\ast}{\longrightarrow} \tilde K(X \vee Y) \,. </annotation></semantics></math></div> <p>By example <a class="maruku-ref" href="#FiniteWedgePropertyForReducedTopologicalKTheory"></a> the two terms involving reduced topological K-theory of a wedge sum are direct sums of the reduced K-theory of the wedge summands:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>Σ</mi><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde K(\Sigma(X \times Y)) \overset{\Sigma i^\ast}{\longrightarrow} \tilde K(\Sigma X) \oplus \tilde K(\Sigma Y) \longrightarrow \tilde K( X \wedge Y ) \longrightarrow \tilde K( X \times Y ) \overset{i^\ast}{\longrightarrow} \tilde{K}(X) \oplus \tilde K(Y) \,. </annotation></semantics></math></div> <p>Now observe that, via example <a class="maruku-ref" href="#FiniteWedgePropertyForReducedTopologicalKTheory"></a>, the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">i^\ast</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><msup><mi>i</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\Sigma i^\ast</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/split+epimorphisms">split epimorphisms</a>, with <a class="existingWikiWord" href="/nlab/show/section">section</a> given by “external direct sum”</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><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>E</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><msubsup><mi>p</mi> <mi>X</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>p</mi> <mi>Y</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde{K}(X) \oplus \tilde K(Y) &\longrightarrow& \tilde K(X \times Y) \\ (E_X, E_Y) &\mapsto& p_X^\ast(E_X) + p_Y^\ast(E_Y) } \,. </annotation></semantics></math></div> <p>This means that the long exact sequence decomposes into <a class="existingWikiWord" href="/nlab/show/short+exact+sequences">short exact sequences</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to \tilde K(X \wedge Y) \longrightarrow \tilde K(X \times Y) \overset{i^\ast}{\longrightarrow} \tilde{K}(X) \oplus \tilde K(Y) \to 0 </annotation></semantics></math></div> <p>which moreover are <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact</a>. This yields the claim.</p> </div> <p>It follows that:</p> <div class="num_prop" id="ExternalTensorProductOnReducedKGroups"> <h6 id="proposition_9">Proposition</h6> <p><strong>(external product on reduced K-groups)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a>. Then the external product on K-groups (def. <a class="maruku-ref" href="#ExternalTensorProductInKTheory"></a>) restricts to <a class="existingWikiWord" href="/nlab/show/reduced+K-groups">reduced K-groups</a> under the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>K</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde K(-) \hookrightarrow K(-)</annotation></semantics></math> from prop. <a class="maruku-ref" href="#KGrupDirectSummandReducedKGroup"></a> and the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>∧</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>K</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>×</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde K(-\wedge -) \hookrightarrow K(-\times -)</annotation></semantics></math> from prop. <a class="maruku-ref" href="#ReducedKTheoryOfProductSpace"></a>, in that there is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \boxtimes</annotation></semantics></math> that makes the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commute</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><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>K</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>⊠</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde{K}(X) \oplus \tilde K(Y) &\hookrightarrow& K(X) \oplus K(Y) \\ {}^{\mathllap{\tilde \boxtimes}}\downarrow && \downarrow^{\mathrlap{\boxtimes}} \\ \tilde K(X \wedge Y) &\hookrightarrow& K(X \times Y) } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>By prop. <a class="maruku-ref" href="#KGrupDirectSummandReducedKGroup"></a> the elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde K(Y)</annotation></semantics></math> are represented by <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundles">virtual vector bundles</a> which vanish when restricted to the base points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">y \in Y</annotation></semantics></math>, respectively. But this implies that their <a class="existingWikiWord" href="/nlab/show/external+tensor+product+of+vector+bundles">external tensor product of vector bundles</a> vanishes over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X \times \{y\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\{x\} \times Y</annotation></semantics></math>. From the proof of prop. <a class="maruku-ref" href="#ReducedKTheoryOfProductSpace"></a> it is the restriction of the product to to these subspaces that gives the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(X \times Y) \simeq \tilde K(X \times Y) \oplus \tilde{K}(X) \oplus \tilde K(Y) \longrightarrow \tilde{K}(X) \oplus \tilde K(Y) </annotation></semantics></math></div> <p>and hence on these element this component vanishes.</p> </div> <h3 id="fundamental_product_theorem">Fundamental product theorem</h3> <p>In order to compute K-classes, one needs the computation of some basic cases, such as that of the K-theory groups of <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a> and of <a class="existingWikiWord" href="/nlab/show/product+spaces">product spaces</a> with <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>-spheres. The <em><a class="existingWikiWord" href="/nlab/show/fundamental+product+theorem+in+K-theory">fundamental product theorem in K-theory</a></em> determines these K-theory groups. Its result is most succinctly summarized by the statement of <em><a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a></em>, to which we turn <a href="#BottPeriodicities">below</a>.</p> <p>Before discussing the product theorem, it is useful to recall the analogous situation in <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^\bullet(-) \coloneqq H^\bullet(\mathbb{Z})</annotation></semantics></math>. Here it is immediate to determine the cohomology groups of the <a class="existingWikiWord" href="/nlab/show/n-spheres">n-spheres</a>, in particular one finds that for the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mi>ℤ</mi><mo stretchy="false">⟨</mo><mi>e</mi><mo stretchy="false">⟩</mo><mo>⊕</mo><mi>ℤ</mi><mo stretchy="false">⟨</mo><mi>h</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">H^\bullet(S^2) = \mathbb{Z}\langle e\rangle \oplus \mathbb{Z}\langle h\rangle </annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><msup><mover><mi>H</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h \in \tilde H^2(S^2)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> of the <a class="existingWikiWord" href="/nlab/show/basic+complex+line+bundle+on+the+2-sphere">basic complex line bundle on the 2-sphere</a>. As a ring this has the trivial product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>h</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">h^2 = 0</annotation></semantics></math>, since by degree-reasons the <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a> goes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">H^2(S^2) \otimes H^2(S^2) \to H^4(S^2) = 0</annotation></semantics></math>.</p> <p>Therefore me may write the ordinary <a class="existingWikiWord" href="/nlab/show/cohomology+ring">cohomology ring</a> of the 2-sphere as the following <a class="existingWikiWord" href="/nlab/show/quotient+ring">quotient ring</a> of the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> in the generator <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>h</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>h</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^\bullet(S^2) \simeq \mathbb{Z}[h]/\left( (h)^2 \right) \,. </annotation></semantics></math></div> <p>Notice that in ordinary cohomology <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> is also the generator of the <a class="existingWikiWord" href="/nlab/show/reduced+cohomology">reduced cohomology</a> group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>H</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">⟨</mo><mi>h</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\tilde H^\bullet(S^2) \simeq \mathbb{Z}\langle h\rangle</annotation></semantics></math>. Now as an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K_{\mathbb{C}}(S^2)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/basic+complex+line+bundle+on+the+2-sphere">basic complex line bundle on the 2-sphere</a> is not reduced, but its image in <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> is the <a class="existingWikiWord" href="/nlab/show/Bott+element">Bott element</a> <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><mi>h</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\beta = h-1</annotation></semantics></math> (def. <a class="maruku-ref" href="#BottElement"></a>). The <a class="existingWikiWord" href="/nlab/show/fundamental+product+theorem+in+topological+K-theory">fundamental product theorem in topological K-theory</a> says, in particular, that the complex topological K-theory of the 2-sphere behaves in just the same way as the ordinary cohomology, if only one replaces the generator <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> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>=</mo><mi>h</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\beta = h-1</annotation></semantics></math>.</p> <p>First of all, the Bott element also squares to zero:</p> <div class="num_prop" id="BottElementNilotentcy"> <h6 id="proposition_10">Proposition</h6> <p><strong>(nilpotency of the <a class="existingWikiWord" href="/nlab/show/Bott+element">Bott element</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>⊂</mo><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">S^2 \subset \mathbb{R}^3</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> with its <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>∈</mo><msub><mi>Vect</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub></mrow><annotation encoding="application/x-tex">h \in Vect_{\mathbb{C}}(S^2)_{/\sim}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/basic+complex+line+bundle+on+the+2-sphere">basic complex line bundle on the 2-sphere</a>. Its image in the <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(S^2)</annotation></semantics></math> satisfies the relation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>h</mi><mo>=</mo><msup><mi>h</mi> <mn>2</mn></msup><mo>+</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 2 h = h^2 + 1 \;\;\Leftrightarrow\;\; (h-1)^2 = 0 </annotation></semantics></math></div></div> <p>A <strong>proof</strong> of this may be obtained by analysis of the relevant <a class="existingWikiWord" href="/nlab/show/clutching+function">clutching function</a>, see <em><a href="basic+complex+line+bundle+on+the+2-sphere#TensorRelationForBasicLineBundleOn2Sphere">here</a></em>.</p> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">h-1</annotation></semantics></math> is the image 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> in the <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> under the splitting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">K(X) \simeq \tilde{K}(X) \oplus \mathbb{Z}</annotation></semantics></math> (by <a href="topological+K-theory#KGrupDirectSummandReducedKGroup">this prop.</a>). This element</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><mn>1</mn><mo>∈</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> h - 1 \in \tilde K_{\mathbb{C}}(S^2) </annotation></semantics></math></div> <p>is the <em><a class="existingWikiWord" href="/nlab/show/Bott+element">Bott element</a></em> of complex <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> (def. <a class="maruku-ref" href="#BottElement"></a>).</p> <p>It follows from prop. <a class="maruku-ref" href="#BottElementNilotentcy"></a> that there is a <a class="existingWikiWord" href="/nlab/show/ring+homomorphism">ring homomorphism</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>h</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>)</mo></mrow></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>h</mi></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mi>h</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z}[h]/\left( (h-1)^2 \right) &\overset{}{\longrightarrow}& K(S^2) \\ h &\overset{\phantom{AAA}}{\mapsto}& h } </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/polynomial+ring">polynomial ring</a> in one abstract generator, <a class="existingWikiWord" href="/nlab/show/quotient+ring">quotiented</a> by this relation, to the <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> ring.</p> <p>More generally, 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/topological+space">topological space</a>, then this induces the composite ring homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>h</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mrow><mo>(</mo><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>)</mo></mrow></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>⊠</mo></mover></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>H</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mo stretchy="false">(</mo><msubsup><mi>π</mi> <mi>X</mi> <mo>*</mo></msubsup><mi>E</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msubsup><mi>π</mi> <mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow> <mo>*</mo></msubsup><mi>H</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ K(X) \otimes \mathbb{Z}[h]/\left((h-1)^2 \right) & \longrightarrow & K(X) \otimes K(S^2) & \overset{\boxtimes}{\longrightarrow} & K(X \times S^2) \\ (E, h) &\overset{\phantom{AAA} }{\mapsto}& (E,H) &\overset{\phantom{AAA}}{\mapsto}& (\pi_{X}^\ast E) \cdot (\pi_{S^2}^\ast H) } </annotation></semantics></math></div> <p>to the topological K-theory ring of the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">X \times S^2</annotation></semantics></math>, where the second map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊠</mo></mrow><annotation encoding="application/x-tex">\boxtimes</annotation></semantics></math> is the external product from def. <a class="maruku-ref" href="#ExternalTensorProductInKTheory"></a>.</p> <div class="num_prop" id="FundamentalProductTheorem"> <h6 id="proposition_11">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/fundamental+product+theorem+in+topological+K-theory">fundamental product theorem in topological K-theory</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>, then this ring homomorphism is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <p>(e.g. <a href="#Hatcher">Hatcher, theorem 2.2</a>)</p> <div class="num_remark"> <h6 id="remark_8">Remark</h6> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">L\to X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex line bundle</a> with class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>l</mi><mo>∈</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">l \in K(X)</annotation></semantics></math> and with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo stretchy="false">(</mo><mn>1</mn><mo>⊕</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P(1 \oplus L)</annotation></semantics></math> denoting its <a class="existingWikiWord" href="/nlab/show/projective+bundle">projective bundle</a> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mi>h</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>l</mi><mo>⋅</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mn>1</mn><mo>⊕</mo><mi>L</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(X)[h]/((h-1)(l \cdot h -1)) \simeq K(P(1 \oplus L)) </annotation></semantics></math></div></div> <p>(e.g. <a href="#Wirthmuller12">Wirthmuller 12, p. 17</a>)</p> <p>As a special case this implies the first statement above:</p> <p>For <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 = \ast</annotation></semantics></math> the product theorem prop. <a class="maruku-ref" href="#FundamentalProductTheorem"></a> says in particular that the first of the two morphisms in the composite is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> (example <a class="maruku-ref" href="#TopologicalKTheoryRingOfThe2Sphere"></a> below) and hence by the <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a>-property for <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> it follows that:</p> <div class="num_cor" id="ExternalProductTheorem"> <h6 id="corollary_2">Corollary</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/external+product+theorem+in+topological+K-theory">external product theorem in topological K-theory</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> we have that the external product in K-theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊠</mo></mrow><annotation encoding="application/x-tex">\boxtimes</annotation></semantics></math> (def. <a class="maruku-ref" href="#ExternalTensorProductInKTheory"></a>) with vector bundles on the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⊠</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \boxtimes \;\colon\; K(X) \otimes K(S^2) \overset{\simeq}{\longrightarrow} K(X \times S^2) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>.</p> </div> <h3 id="BottPeriodicities">Bott periodicity</h3> <p>When restricted to <a class="existingWikiWord" href="/nlab/show/reduced+K-theory">reduced K-theory</a> then the external product theorem (cor. <a class="maruku-ref" href="#ExternalProductTheorem"></a>) yields the statement of <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a> of topological K-theory:</p> <div class="num_prop" id="BottPeriodicity"> <h6 id="proposition_12">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a>)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>.</p> <p>Then the external product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde X</annotation></semantics></math> in reduced K-theory (prop. <a class="maruku-ref" href="#ExternalTensorProductInKTheory"></a>) with the image of the <a class="existingWikiWord" href="/nlab/show/basic+complex+line+bundle+on+the+2-sphere">basic complex line bundle on the 2-sphere</a> in reduced K-theory yields an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/reduced+K-groups">reduced K-groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mn>2</mn></msup><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (h-1) \tilde \boxtimes (-) \;\colon\; \tilde{K}(X) \overset{\simeq}{\longrightarrow} \tilde K(\Sigma^2 X) </annotation></semantics></math></div> <p>from that 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 that of its double <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mn>2</mn></msup><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma^2 X</annotation></semantics></math>.</p> </div> <p><a href="#Wirthmuller12">e.g. Wirthmuller 12, p. 34 (36 of 67)</a></p> <div class="proof"> <h6 id="proof_12">Proof</h6> <p>By <a href="topologica+K-theory#ReducedKTheoryOfProductSpace">this example</a> there is for any two pointed compact Hausdorff spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo>∧</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde K(Y \times X) \simeq \tilde K(Y \wedge X) \oplus \tilde K(Y) \oplus \tilde{K}(X) </annotation></semantics></math></div> <p>relating the reduced K-theory of the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> with that of the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a>.</p> <p>Using this and the fact that for any pointed compact Hausdorff space <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> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">K(Z) \simeq \tilde K(Z) \oplus \mathbb{Z}</annotation></semantics></math> (<a href="topological+K-theory#KGrupDirectSummandReducedKGroup">this prop.</a>) the isomorphism of the external product theorem (cor. <a class="maruku-ref" href="#ExternalProductTheorem"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>⊗</mo><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mo>⊠</mo></munderover><mi>K</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K(S^2) \otimes K(X) \underoverset{\simeq}{\boxtimes}{\longrightarrow} K(S^2 \times X) </annotation></semantics></math></div> <p>becomes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>⊕</mo><mi>ℤ</mi><mo>)</mo></mrow><mo>⊗</mo><mrow><mo>(</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>ℤ</mi><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>ℤ</mi><mo>)</mo></mrow><mo>≃</mo><mrow><mo>(</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>∧</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>ℤ</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \tilde K(S^2) \oplus \mathbb{Z} \right) \otimes \left( \tilde{K}(X) \oplus \mathbb{Z} \right) \;\simeq\; \left( \tilde K(S^2 \times X) \oplus \mathbb{Z} \right) \simeq \left( \tilde K(S^2 \wedge X) \oplus \tilde K(S^2) \oplus \tilde{K}(X) \oplus \mathbb{Z} \right) \,. </annotation></semantics></math></div> <p>Multiplying out and chasing through the constructions to see that this reduces to an isomorphism on the common summand <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\tilde K(S^2) \oplus \tilde{K}(X) \oplus \mathbb{Z}</annotation></semantics></math>, this yields an isomorphism of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover></munderover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>∧</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mn>2</mn></msup><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \tilde K(S^2) \otimes \tilde{K}(X) \underoverset{\simeq}{\tilde \boxtimes}{\longrightarrow} \tilde K(S^2 \wedge X) = \tilde K(\Sigma^2 X) \,, </annotation></semantics></math></div> <p>where on the right we used that <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> with the 2-sphere is the same as double <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a>.</p> <p>Finally there is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</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><mi>ℤ</mi></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mi>β</mi></munderover></mtd> <mtd><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z} &\underoverset{\simeq}{ \beta }{\longrightarrow}& \tilde K_{\mathbb{C}}(S^2) \\ 1 &\overset{\phantom{AAA}}{\mapsto}& (h-1) } </annotation></semantics></math></div> <p>(example <a class="maruku-ref" href="#TopologicalKTheoryRingOfThe2Sphere"></a>). The composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>ℤ</mi><mo>⊗</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>β</mi><mo>⊗</mo><mi>id</mi></mrow></mover><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>⊗</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover></munderover></mtd> <mtd><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>∧</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mn>2</mn></msup><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>E</mi><mo>−</mo><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAAA</mi></mphantom></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>E</mi><mo>−</mo><msub><mi>rk</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \tilde K_{\mathbb{C}}(X) & \simeq \mathbb{Z} \otimes \tilde K_{\mathbb{C}}(X) \overset{ \beta \otimes id }{\longrightarrow} \tilde K_{\mathbb{C}}(S^2) \otimes \tilde K_{\mathbb{C}}(X) \underoverset{\simeq}{\tilde \boxtimes}{\longrightarrow} & \tilde K_{\mathbb{C}}(S^2 \wedge X) = \tilde K_{\mathbb{C}}(\Sigma^2 X) \\ E - rk_x(E) &\overset{\phantom{AAAA}}{\mapsto}& (h-1) \tilde \boxtimes (E - rk_x(E)) } </annotation></semantics></math></div> <p>is the isomorphism to be established.</p> </div> <h3 id="GradedRingStructure">Graded-commutative ring structure</h3> <p>The external product on reduced K-groups from prop. <a class="maruku-ref" href="#ExternalTensorProductOnReducedKGroups"></a> allows to extend the <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> structure from the plain K-groups (remark <a class="maruku-ref" href="#KTheoryRing"></a>) to a ring structure on the graded K-groups from def. <a class="maruku-ref" href="#GradedKGroups"></a>. This is def. <a class="maruku-ref" href="#ProductOnGradedKGroups"></a> below.</p> <p>To state this definition, recall that</p> <ol> <li> <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/pointed+topological+space">pointed topological space</a> then the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> map to its <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \times X</annotation></semantics></math> induced a diagonal to the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∧</mo><mi>X</mi><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∨</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \wedge X = (X \times X)/(X \vee 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><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>Δ</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi><mover><mo>⟶</mo><mi>q</mi></mover><mi>X</mi><mo>∧</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \overset{\Delta_X}{\longrightarrow} X \times X \overset{q}{\longrightarrow} X\wedge X </annotation></semantics></math></div></li> <li> <p>since <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a> is equivalently <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> with the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mi>X</mi><mo>≃</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma X \simeq S^1 \wedge X</annotation></semantics></math>, there are induced “partial diagonal maps” of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>∘</mo><msub><mi>Δ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Σ</mi><mi>X</mi><mo>≃</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mi>X</mi><mover><mo>⟶</mo><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mo stretchy="false">(</mo><mi>q</mi><mo>∘</mo><msub><mi>Δ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow></mover><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mi>X</mi><mo>∧</mo><mi>X</mi><mo>≃</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>∧</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \Sigma (q \circ \Delta_X) \;\colon\; \Sigma X \simeq S^1 \wedge X \overset{S^1 \wedge (q\circ \Delta_X)}{\longrightarrow} S^1 \wedge X \wedge X \simeq (\Sigma X) \wedge X </annotation></semantics></math></div> <p>etc.</p> </li> </ol> <div class="num_defn" id="ProductOnGradedKGroups"> <h6 id="definition_13">Definition</h6> <p><strong>(product on graded K-groups)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>, the <em>product on graded K-groups</em></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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><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><msup><mi>K</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>K</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (-)\cdot (-) \;\colon\; K^\bullet(X) \otimes K^\bullet(X) \longrightarrow K^\bullet(X) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> which on the direct summands <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde K^0(X) \coloneqq \tilde{K}(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde K^1(X) \coloneqq \tilde K(\Sigma X)</annotation></semantics></math> is given by the following morphisms, which are <a class="existingWikiWord" href="/nlab/show/composition">composites</a> of the external product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde \boxtimes</annotation></semantics></math> on reduced K-groups from prop. <a class="maruku-ref" href="#ExternalTensorProductOnReducedKGroups"></a> with pullbacks along the above suspended diagonal maps:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde{K}(X) \otimes \tilde{K}(X) \overset{\tilde \boxtimes}{\longrightarrow} \tilde{K}(X) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">(</mo><mi>q</mi><mo>∘</mo><msub><mi>Δ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde{K}(X) \otimes \tilde K(\Sigma X) \overset{\tilde \boxtimes}{\longrightarrow} \tilde K(X \wedge (\Sigma X)) \overset{ (\Sigma(q \circ \Delta_X))^\ast }{\longrightarrow} \tilde K(\Sigma X) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mover><mo>⊠</mo><mo stretchy="false">˜</mo></mover></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>∧</mo><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>q</mi><mo>∘</mo><msub><mi>Δ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mover><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>Σ</mi> <mn>2</mn></msup><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \tilde K(\Sigma X) \otimes \tilde K(\Sigma X) \overset{\tilde \boxtimes}{\longrightarrow} \tilde K( (\Sigma X) \wedge (\Sigma X)) \overset{ (\Sigma^2(q \circ \Delta_X))^\ast }{\longrightarrow} \tilde K(\Sigma^2 X) \simeq \tilde{K}(X) \,, </annotation></semantics></math></div> <p>where the last isomorphism on the right is <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a> isomorphism (prop. <a class="maruku-ref" href="#BottPeriodicity"></a>).</p> </div> <h3 id="ClassifyingSpace">Classifying space</h3> <p>We discuss how the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\tilde K^0</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/stable+unitary+group">stable unitary group</a>.</p> <div class="num_defn" id="BUn"> <h6 id="definition_14">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of the <a class="existingWikiWord" href="/nlab/show/stable+unitary+group">stable unitary group</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> in dimension <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> in dimension <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>, both regarded as <a class="existingWikiWord" href="/nlab/show/topological+groups">topological groups</a> in the standard way. Write <a class="existingWikiWord" href="/nlab/show/BU%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B U(n)</annotation> </semantics> </math></a>, <a class="existingWikiWord" href="/nlab/show/BO%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B O(n)</annotation> </semantics> </math></a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∈</mo></mrow><annotation encoding="application/x-tex">\in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> for the corresponding <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a>.</p> <p>Write</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>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>:</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [X, B O(n)] := \pi_0 Top(X, B O(n)) </annotation></semantics></math></div> <p>and</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>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>:</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [X, B U(n)] := \pi_0 Top(X, B U(n)) </annotation></semantics></math></div> <p>for the set of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>-classes of <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to B U(n)</annotation></semantics></math>.</p> </div> <div class="num_prop" id="BUnClassifyingSpace"> <h6 id="proposition_13">Proposition</h6> <p>This is equivalently the set of <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> classes of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>- or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s 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> as well as of rank-<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> real or complex <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s 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>, respectively:</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>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≃</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Vect</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [X, B O(n)] \simeq O(n) Bund(X) \simeq Vect_{\mathbb{R}}(X,n) \,, </annotation></semantics></math></div><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>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Vect</mi> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X, B U(n)] \simeq U(n) Bund(X) \simeq Vect_{\mathbb{C}}(X,n) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="InclusionOfUns"> <h6 id="definition_15">Definition</h6> <p>For each <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> let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> U(n) \to U(n+1) </annotation></semantics></math></div> <p>be the inclusion of <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>s given by inclusion of <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 \times n</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/matrices">matrices</a> into <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><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1) \times (n+1)</annotation></semantics></math>-matrices given by the block-diagonal form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mi>g</mi><mo>]</mo></mrow><mo>↦</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mn>1</mn></mtd> <mtd><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>g</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left[g\right] \mapsto \left[ \array{ 1 & [0] \\ [0] & [g] } \right] \,. </annotation></semantics></math></div> <p>This induces a corresponding sequence of morphisms of classifying spaces, def. <a class="maruku-ref" href="#BUn"></a>, in <a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>↪</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> B U(0) \hookrightarrow B U(1) \hookrightarrow B U(2) \hookrightarrow \cdots \,. </annotation></semantics></math></div> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo>:</mo><mo>=</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> B U := {\lim_{\to}}_{n \in \mathbb{N}} B U(n) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a> (the “homotopy <a class="existingWikiWord" href="/nlab/show/direct+limit">direct limit</a>”) over this diagram (see at <em><a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></em> the section <em><a href="homotopy+limit#SequentialHocolims">Sequential homotopy colimits</a></em>).</p> </div> <div class="num_remark"> <h6 id="remark_9">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">B U</annotation></semantics></math> is <strong>not</strong> equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B U(\mathcal{H}) </annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathcal{H})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> on a separable infinite-dimensional <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><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>. In fact the latter is <a class="existingWikiWord" href="/nlab/show/contractible">contractible</a>, hence has a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> to the point</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> B U(\mathcal{H}) \simeq * </annotation></semantics></math></div> <p>while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">B U</annotation></semantics></math> has nontrivial <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s in arbitrary higher degree (by <a class="existingWikiWord" href="/nlab/show/Kuiper%27s+theorem">Kuiper's theorem</a>).</p> <p>But there is the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><msub><mo stretchy="false">)</mo> <mi>𝒦</mi></msub><mo>⊂</mo><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathcal{H})_{\mathcal{K}} \subset U(\mathcal{H})</annotation></semantics></math> of unitary operators that differ from the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> by a <a class="existingWikiWord" href="/nlab/show/compact+operator">compact operator</a>. This is essentially <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>=</mo><mi>Ω</mi><mi>B</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">U = \Omega B U</annotation></semantics></math>. See <a href="#Uk">below</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_14">Proposition</h6> <p>Write <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">\mathbb{Z}</annotation></semantics></math> for the set of <a class="existingWikiWord" href="/nlab/show/integer">integer</a>s regarded as a <a class="existingWikiWord" href="/nlab/show/discrete+topological+space">discrete topological space</a>.</p> <p>The product spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>O</mi><mo>×</mo><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>B</mi><mi>U</mi><mo>×</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> B O \times \mathbb{Z}\,,\;\;\;\;\;B U \times \mathbb{Z} </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> for real and complex <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>-theory, respectively: for every compact Hausdorff topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, we have an isomorphism of groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>U</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde{K}(X) \simeq [X, B U ] \,. </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>U</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K(X) \simeq [X, B U \times \mathbb{Z}] \,. </annotation></semantics></math></div></div> <p>See for instance (<a href="#Friedlander">Friedlander, prop. 3.2</a>) or (<a href="#Karoubi">Karoubi II, prop. 1.32, theorem 1.33</a>).</p> <div class="proof"> <h6 id="proof_13">Proof</h6> <p>First consider the statement for reduced cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde{K}(X)</annotation></semantics></math>:</p> <p>Since a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact topological space</a> is a <a class="existingWikiWord" href="/nlab/show/compact+object">compact object</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> (and using that the <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B U(n)</annotation></semantics></math> are (see there) <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact topological space</a>s, hence normal, and since the inclusion morphisms are closed inclusions (…)) the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> out of it commutes with the <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>n</mi></msub><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mrow><munder><mi>lim</mi> <mo>→</mo></munder></mrow> <mi>n</mi></msub><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} Top(X, B U) &= Top(X, {\lim_\to}_n B U(n)) \\ & \simeq {\lim_\to}_n Top(X, B U (n)) \end{aligned} \,. </annotation></semantics></math></div> <p>Since <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>B</mi><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[X, B U(n)] \simeq U(n) Bund(X)</annotation></semantics></math>, in the last line the colimit is over <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s of all ranks and identifies two if they become isomorphic after adding a trivial bundle of some finite rank.</p> <p>For the full statement use that by prop. <a class="maruku-ref" href="#missing"></a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K(X) \simeq H^0(X, \mathbb{Z}) \oplus \tilde{K}(X) \,. </annotation></semantics></math></div> <p>Because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">H^0(X,\mathbb{Z}) \simeq [X, \mathbb{Z}]</annotation></semantics></math> it follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>⊕</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">]</mo><mo>×</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>U</mi><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>U</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^0(X, \mathbb{Z}) \oplus \tilde{K}(X) \simeq [X, \mathbb{Z}] \times [X, B U] \simeq [X, B U \times \mathbb{Z}] \,. </annotation></semantics></math></div></div> <p> <div class='num_remark'> <h6>Remark</h6> <p>In this sense, topological <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>-theory may be regarded as the <em>stabilization</em> of the <a class="existingWikiWord" href="/nlab/show/unstable+topological+K-theory">unstable topological K-theory</a> groups <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>BU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,BU(n)]</annotation></semantics></math> and <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>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,U(n)]</annotation></semantics></math>.</p> </div> </p> <p>There is another variant on the classifying space</p> <div class="num_defn" id="Uk"> <h6 id="definition_16">Definition</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>𝒦</mi></msub><mo>=</mo><mrow><mo>{</mo><mi>g</mi><mo>∈</mo><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>g</mi><mo>−</mo><mi>id</mi><mo>∈</mo><mi>𝒦</mi><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> U_{\mathcal{K}} = \left\{ g \in U(\mathcal{H}) | g - id \in \mathcal{K} \right\} </annotation></semantics></math></div> <p>be the group of unitary operators on a <a class="existingWikiWord" href="/nlab/show/separable+Hilbert+space">separable Hilbert space</a> <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">\mathcal{H}</annotation></semantics></math> which differ from the identity by a <a class="existingWikiWord" href="/nlab/show/compact+operator">compact operator</a>.</p> </div> <p>Palais showed that</p> <div class="num_prop"> <h6 id="proposition_15">Proposition</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>𝒦</mi></msub></mrow><annotation encoding="application/x-tex">U_\mathcal{K}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalent</a> model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">B U</annotation></semantics></math>. It is in fact the <a class="existingWikiWord" href="/nlab/show/norm+closure">norm closure</a> of the evident model of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi></mrow><annotation encoding="application/x-tex">B U</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathcal{H})</annotation></semantics></math>.</p> <p>Moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>𝒦</mi></msub><mo>⊂</mo><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_{\mathcal{K}} \subset U(\mathcal{H})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Banach+Lie+group">Banach Lie</a> <a class="existingWikiWord" href="/nlab/show/normal+subgroup">normal subgroup</a>.</p> </div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(\mathcal{H})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/contractible">contractible</a>, it follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mi>U</mi> <mi>𝒦</mi></msub><mo>≔</mo><mi>U</mi><mo stretchy="false">(</mo><mi>ℋ</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>U</mi> <mi>𝒦</mi></msub></mrow><annotation encoding="application/x-tex"> B U_{\mathcal{K}} \coloneqq U(\mathcal{H})/U_{\mathcal{K}} </annotation></semantics></math></div> <p>is a model for the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of reduced K-theory.</p> <h3 id="of_noncompact_spaces">Of non-compact spaces</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a compact Lie group with <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">B G</annotation></semantics></math> (in general non-compact) then the map from the <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>Grp</mi><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mo>∼</mo></msub><mo>,</mo><mo>⊕</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{K}(B G) \coloneqq Grp(Vect(B G)/_\sim, \oplus)</annotation></semantics></math> (def. <a class="maruku-ref" href="#GrothendieckGroupKTheory"></a>) to the representable K-theory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>rep</mi></msub><mo>≔</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>U</mi><mo>×</mo><mi>ℤ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">K(B G)_{rep} \coloneqq [X, B U \times\mathbb{Z}]</annotation></semantics></math> (def. <a class="maruku-ref" href="#RepresntableTopologicalKTheory"></a>) is <a class="existingWikiWord" href="/nlab/show/injective+function">injective</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕂</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>K</mi><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>rep</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{K}(B G) \hookrightarrow K(B G)_{rep} \,. </annotation></semantics></math></div> <p><a href="#JackowskiOliver">Jackowski-Oliver</a></p> <h3 id="AsAGeneralizedCohomologyTheory">As a generalized cohomology theory</h3> <p>Topological K-theory satisfies the axioms of a <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology+theory">generalized (Eilenberg-Steenrod) cohomology theory</a> (<a href="#AtiyahHirzebruch61">Atiyah-Hirzebruch 61, 1.8</a>).</p> <p>This is essentially the statement of the long exact sequences <a href="#ExactSequences">above</a>.</p> <h3 id="ComplexOrientationAndFormalGroupLaw">Complex orientation and formal group law</h3> <p>A <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative</a> <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theory <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> is called <em><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology">complex orientable</a></em> if the element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>E</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>≃</mo><mover><mi>E</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">1 \in E(\ast) \simeq \tilde E(S^0)</annotation></semantics></math> is in the image of the pullback morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>i</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde i^\ast \;\colon\; \tilde E^2(B U(1)) \longrightarrow \tilde E^2(S^2) \simeq \tilde E^0(S^0) \,. </annotation></semantics></math></div> <p>If so, then a choice of pre-image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>c</mi> <mn>1</mn> <mi>E</mi></msubsup><mo>∈</mo><msup><mi>E</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c^E_1 \in E^2(B U(1))</annotation></semantics></math> is a choice of <em>complex orientation</em> (<a href="complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheory">this def.</a>).</p> <p>Now for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><msub><mi>K</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">E = K_{\mathbb{C}}</annotation></semantics></math> being complex topological K-theory regarded as a generalized cohomology theory as <a href="#AsAGeneralizedCohomologyTheory">above</a>, then by <a class="existingWikiWord" href="/nlab/show/Bott+periodicity">Bott periodicity</a> (prop. <a class="maruku-ref" href="#BottPeriodicity"></a>) and by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℤ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde K_{\mathbb{Z}}(S^2) \simeq \mathbb{Z} \cdot (h-1)</annotation></semantics></math> (example <a class="maruku-ref" href="#ComplexTopologicalKTheoryOfTheCircle"></a>) this reduces to the statement that there is an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>c</mi> <mn>1</mn> <mi>K</mi></msubsup><mo>∈</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c^K_1 \in \tilde K_{\mathbb{C}}(B U(1))</annotation></semantics></math> such that its image under</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>i</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>h</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde i^\ast \;\colon\; \tilde K_{\mathbb{C}}(B U(1)) \longrightarrow \tilde K_{\mathbb{C}}(S^2) \simeq \mathbb{Z} \cdot (h-1) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/Bott+element">Bott element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">h-1</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a> difference between the <a class="existingWikiWord" href="/nlab/show/basic+complex+line+bundle+on+the+2-sphere">basic complex line bundle on the 2-sphere</a> and the <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial</a> <a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex line bundle</a>.</p> <p>By the very nature of the <a class="existingWikiWord" href="/nlab/show/basic+complex+line+bundle+on+the+2-sphere">basic complex line bundle on the 2-sphere</a> <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>, it is the restriction of the <a class="existingWikiWord" href="/nlab/show/universal+complex+line+bundle">universal</a>/<a class="existingWikiWord" href="/nlab/show/tautological+line+bundle">tautological</a> <a class="existingWikiWord" href="/nlab/show/complex+line+bundle">complex line bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(1)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">B U(1) \simeq \mathbb{C}P^\infty</annotation></semantics></math> along the defining cell inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>↪</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>∞</mn></msup><mo>≃</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i \colon S^2 \hookrightarrow \mathbb{C}P^\infty \simeq B U(1)</annotation></semantics></math>. Hence if we set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>c</mi> <mi>K</mi> <mn>1</mn></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>ℂ</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> c_K^1 \;\coloneqq\; \mathcal{O}(1)-1 \; \in \tilde K_{\mathbb{C}}(B U(1)) </annotation></semantics></math></div> <p>then this is a <a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology">complex orientation</a> for complex topological K-theory.</p> <p>From this we obtain the <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a> associated with topological K-theory (from <a href="complex+oriented+cohomology+theory#ComplexOrientedCohomologyTheoryFormalGroupLaw">this prop.</a>):</p> <p>By the nature of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <a class="existingWikiWord" href="/nlab/show/BU%28n%29"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>B</mi> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">B U(1)</annotation> </semantics> </math></a> we have that for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⟶</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mu \;\colon\; B U(1) \times B U(1) \longrightarrow B U(1) </annotation></semantics></math></div> <p>the group product operation, which classifies the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+bundles">tensor product of line bundles</a>, that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mo>*</mo></msup><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>pr</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><msubsup><mi>pr</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mu^\ast \mathcal{O}(1) \simeq pr_1^\ast \mathcal{O}(1) \otimes_{B U(1)} pr_2^\ast \mathcal{O}(1) \,, </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>pr</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> pr_i\colon B U(1) \times B U(1) \to B U(1) </annotation></semantics></math></div> <p>are the two <a class="existingWikiWord" href="/nlab/show/projections">projections</a> out of the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a>. Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msup><mi>μ</mi> <mo>*</mo></msup><msubsup><mi>c</mi> <mn>1</mn> <mi>K</mi></msubsup></mtd> <mtd><mo>≔</mo><msup><mi>μ</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><msubsup><mi>pr</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><msubsup><mi>pr</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><msubsup><mi>pr</mi> <mn>1</mn> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><msubsup><mi>pr</mi> <mn>2</mn> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>+</mo><msubsup><mi>pr</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>pr</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>𝒪</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><msubsup><mi>pr</mi> <mn>1</mn> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><msubsup><mi>pr</mi> <mn>2</mn> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><msubsup><mi>pr</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><msubsup><mi>pr</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mu^\ast c_1^K &\coloneqq \mu^\ast (\mathcal{O}(1) - 1) \\ & = \left(pr_1^\ast \mathcal{O}(1)\right) \cdot \left(pr_2^\ast \mathcal{O}(1)\right) - 1 \\ & = \left(pr_1^\ast (\mathcal{O}(1) -1)\right) \cdot \left(pr_2^\ast( \mathcal{O}(1) -1)\right) + pr_1^\ast \mathcal{O}(1) + pr_2^\ast \mathcal{O}(2) - 2 \\ & = \left(pr_1^\ast (\mathcal{O}(1) -1)\right) \cdot \left(pr_2^\ast( \mathcal{O}(1) -1)\right) + \left(pr_1^\ast \mathcal{O}(1) - 1\right) + \left(pr_2^\ast \mathcal{O}(1) - 1\right) \end{aligned} </annotation></semantics></math></div> <p>This shows that the <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a> associated with the complex orientation of complex topological K-theory is that of the <em><a class="existingWikiWord" href="/nlab/show/formal+multiplicative+group">formal multiplicative group</a></em> given by</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 stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>x</mi><mi>y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f(x,y) = x + y + x y \,. </annotation></semantics></math></div> <h3 id="spectrum">Spectrum</h3> <p>Being a <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> theory, topological K-theory is represented by a <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a>: the <em><a class="existingWikiWord" href="/nlab/show/K-theory+spectrum">K-theory spectrum</a></em>.</p> <p>e.g. <a href="#Switzer75">Switzer 75, p. 216</a></p> <p>The degree-0 part of this spectrum, i.e. the classifying space for degree 0 topological <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>-theory is modeled in particular by the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fred</mi></mrow><annotation encoding="application/x-tex">Fred</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Fredholm+operator">Fredholm operator</a>s.</p> <h3 id="ring_spectrum">Ring spectrum</h3> <p>This <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> spectrum has the structure of a <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a></p> <p>(e.g. <a href="#Switzer75">Switzer 75, section 13.90, around p. 300</a>,</p> <p>see also p. 205 (213 of 251) in <em><a class="existingWikiWord" href="/nlab/show/A+Concise+Course+in+Algebraic+Topology">A Concise Course in Algebraic Topology</a></em>)</p> <p>(…)</p> <h3 id="chromatic_filtration">Chromatic filtration</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/chromatic+homotopy+theory">chromatic homotopy theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a></th><th><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented cohomology theory</a></th><th><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>/<a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a></th><th><a class="existingWikiWord" href="/nlab/show/real+oriented+cohomology+theory">real oriented cohomology theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H \mathbb{Z}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/HZR-theory">HZR-theory</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">0th <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(0)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory+spectrum">complex K-theory spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KU</mi></mrow><annotation encoding="application/x-tex">KU</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">first <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(1)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">first <a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(1)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+spectrum">elliptic spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ell</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">Ell_E</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">second <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(2)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">second <a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(2)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> of <a class="existingWikiWord" href="/nlab/show/KU">KU</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(KU)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">3 …10</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+cohomology">K3 cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/K3+spectrum">K3 spectrum</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><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/Morava+K-theory">Morava K-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(n)</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><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>th <a class="existingWikiWord" href="/nlab/show/Morava+E-theory">Morava E-theory</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E(n)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BPR-theory">BPR-theory</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a> applied to chrom. level <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></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(E_n)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/red-shift+conjecture">red-shift conjecture</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><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+cobordism+cohomology">complex cobordism cohomology</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/MU">MU</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/MR-theory">MR-theory</a></td></tr> </tbody></table> </div> <h3 id="as_the_shape_of_the_smooth_ktheory_spectrum">As the shape of the smooth K-theory spectrum</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></em>.</p> <h3 id="RelationToAlgebraicKTheory">Relation to algebraic K-theory</h3> <p>The topological K-theory over a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is not identical with the <em><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a></em> of the ring of functions 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>, but the two are closely related. See for instance (<a href="#Paluch">Paluch</a>, <a href="#Rosenberg">Rosenberg</a>). See at <em><a class="existingWikiWord" href="/nlab/show/comparison+map+between+algebraic+and+topological+K-theory">comparison map between algebraic and topological K-theory</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/BU%28n%29">BU(n)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/virtual+vector+bundle">virtual vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><strong>topological K-theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah-Bott-Shapiro+isomorphism">Atiyah-Bott-Shapiro isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+index">topological index</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KO-theory">KO-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vectorial+bundle">vectorial bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unstable+K-theory">unstable K-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+K-theory">groupoid K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</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/twisted+differential+K-theory">twisted differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a>, <a class="existingWikiWord" href="/nlab/show/orbifold+K-theory">orbifold K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+differential+K-theory">equivariant differential K-theory</a>, <a class="existingWikiWord" href="/nlab/show/orbifold+differential+K-theory">orbifold differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-topological+K-theory">semi-topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/iterated+K-theory">iterated K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tate+K-theory">Tate K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/KU-local+stable+homotopy+theory">KU-local stable homotopy theory</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a></th><th><a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> / <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a></th><th><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a> to <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">orientation in generalized cohomology</a></th><th><a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">generalized orientation</a>/<a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></th><th><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></th><th>incarnation as <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> in <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></th></tr></thead><tbody><tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">complex K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KU</mi></mrow><annotation encoding="application/x-tex">KU</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+class">third integral SW class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">W_3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin%E1%B6%9C+structure">spinᶜ structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+quantization+by+push-forward">K-theoretic geometric quantization</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Freed-Witten+anomaly">Freed-Witten anomaly</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/EO%28n%29">EO(n)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Stiefel-Whitney+class">Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">w_4</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+Morava+K-theory">integral Morava K-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde K(2)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/integral+Stiefel-Whitney+class">seventh integral SW class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mn>7</mn></msub></mrow><annotation encoding="application/x-tex">W_7</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Diaconescu-Moore-Witten+anomaly">Diaconescu-Moore-Witten anomaly</a> in <a href="Diaconescu-Moore-Witten+anomaly#ReferencesInterpretationInSecondMoravaKTheory">Kriz-Sati interpretation</a></td></tr> </tbody></table> </div><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohomology+theories">cohomology theories</a> of <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> on <a class="existingWikiWord" href="/nlab/show/orientifolds">orientifolds</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></th><th><a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></th><th><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>-field <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli</a></th><th><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+2-bundle">line 2-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><msup><mi>ℤ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">H\mathbb{Z}^3</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+II+superstring">type II superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Pic(KU)//\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KR-theory">KR-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>KR</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">KR^\bullet</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIA+superstring">type IIA superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B GL_1(KU)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">KU-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>KU</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">KU^0</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+IIB+superstring">type IIB superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B GL_1(KU)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+K-theory">KU-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>KU</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">KU^1</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/type+I+superstring">type I superstring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Pic(KU)//\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KO-theory">KO-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KO</mi></mrow><annotation encoding="application/x-tex">KO</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">type <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>I</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde I</annotation></semantics></math> superstring</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pic</mi><mo stretchy="false">(</mo><mi>KU</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Pic(KU)//\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/KSC-theory">KSC-theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>KSC</mi></mrow><annotation encoding="application/x-tex">KSC</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The “ring of complex vector bundles” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math> was introduced in</p> <ul> <li id="AtiyahHirzebruch61"><a class="existingWikiWord" href="/nlab/show/M.+F.+Atiyah">M. F. Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/F.+Hirzebruch">F. Hirzebruch</a>, <em>Riemann-Roch theorems for differentiable manifolds</em>, Bull. Amer. Math Soc. vol. 65 (1959) pp. 276-281 (<a href="https://projecteuclid.org/euclid.bams/1183523205">euclid:bams/1183523205</a>)</li> </ul> <p>and shown to give a <a class="existingWikiWord" href="/nlab/show/Whitehead-generalized+cohomology">Whitehead-generalized cohomology</a> theory in</p> <ul> <li id="AtiyahHirzebruch61"><a class="existingWikiWord" href="/nlab/show/M.+F.+Atiyah">M. F. Atiyah</a>, <a class="existingWikiWord" href="/nlab/show/F.+Hirzebruch">F. Hirzebruch</a>, <em>Vector bundles and homogeneous spaces</em>, Proc. Sympos. Pure Math. <strong>III</strong>, American Mathematical Society (1961) 7-38 [<a href="https://doi.org/10.1142/9789814401319_0008">doi:10.1142/9789814401319_0008</a>, <a href="http://hirzebruch.mpim-bonn.mpg.de/87/">web</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=0139181">MR 0139181</a>]</li> </ul> <p>Early lecture notes:</p> <ul> <li id="Atiyah67"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>K-theory</em>, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin (1967) [<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/atiyahk.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/AtiyahKTheory.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <em>Lectures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(X)</annotation></semantics></math></em>, Benjamin (1969) [<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/bottk.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Bott-KTheory.pdf" title="pdf">pdf</a>]</p> <p>Russian transl. by B. Yu. Sternin, Matematika <strong>11</strong> 2 (1967) 32–56 [<a href="https://www.mathnet.ru/eng/mat424">mathnet:mat424</a>]</p> </li> </ul> <p>Representable K-theory over non-compact spaces was considered in</p> <ul> <li id="Karoubi70"><a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <em>Espaces Classifiants en K-Théorie</em>, Transactions of the American Mathematical Society Vol. 147, No. 1 (Jan., 1970), pp. 75-115 (<a href="http://www.jstor.org/stable/1995218">jstor</a>)</li> </ul> <p>and (over <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> in the context of <a class="existingWikiWord" href="/nlab/show/equivariant+K-theory">equivariant K-theory</a>) in:</p> <ul> <li id="SegalAtiyah69"><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, section 4 of: <em>Equivariant K-theory and completion</em>, J. Differential Geometry <strong>3</strong> (1969) 1-18 [MR 0259946]</li> </ul> <p>Early lecture notes on topological K-theory in a general context of <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/generalized+cohomology+theory">generalized cohomology theory</a> includes</p> <ul> <li id="Adams74"><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, part III, section 2 of <em><a class="existingWikiWord" href="/nlab/show/Stable+homotopy+and+generalised+homology">Stable homotopy and generalised homology</a></em>, 1974</li> </ul> <p>Textbook accounts:</p> <ul> <li id="ConnerFloyd66"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Conner">Pierre Conner</a>, <a class="existingWikiWord" href="/nlab/show/Edwin+Floyd">Edwin Floyd</a>, <em><a class="existingWikiWord" href="/nlab/show/The+Relation+of+Cobordism+to+K-Theories">The Relation of Cobordism to K-Theories</a></em>, Lecture Notes in Mathematics <strong>28</strong> Springer 1966 (<a href="https://link.springer.com/book/10.1007/BFb0071091">doi:10.1007/BFb0071091</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=216511">MR216511</a>)</p> <blockquote> <p>(in view of <a class="existingWikiWord" href="/nlab/show/U-bordism+theory">U-bordism theory</a> and the <a class="existingWikiWord" href="/nlab/show/e-invariant">e-invariant</a>)</p> </blockquote> </li> <li id="Switzer75"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Switzer">Robert Switzer</a>, sections 11 and 13.90 of: <em>Algebraic Topology - Homotopy and Homology</em>, Grundlehren <strong>212</strong> Springer (1975) [<a href="https://doi.org/10.1007/978-3-642-61923-6_12">doi:10.1007/978-3-642-61923-6_12</a>]</p> </li> <li id="Karoubi"> <p><a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <em>K-Theory – An introduction</em>, Grundlehren der mathematischen Wissenschaften <strong>226</strong>, Springer (1978) [<a href="https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf">pdf</a>, <a href="https://link.springer.com/book/10.1007%2F978-3-540-79890-3">doi:10.1007%2F978-3-540-79890-3</a>]</p> </li> <li id="Hatcher"> <p><a class="existingWikiWord" href="/nlab/show/Allen+Hatcher">Allen Hatcher</a>, <em>Vector bundles and K-theory</em>, 2003/2009 (<a href="http://www.math.cornell.edu/~hatcher/VBKT/VBpage.html">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dai+Tamaki">Dai Tamaki</a>, <a class="existingWikiWord" href="/nlab/show/Akira+Kono">Akira Kono</a>, Section 4.1 in: <em>Generalized Cohomology</em>, Translations of Mathematical Monographs, American Mathematical Society, 2006 (<a href="https://bookstore.ams.org/mmono-230">ISBN: 978-0-8218-3514-2</a>)</p> </li> </ul> <p>Further introductions:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/H.+Blaine+Lawson">H. Blaine Lawson</a>, <a class="existingWikiWord" href="/nlab/show/Marie-Louise+Michelsohn">Marie-Louise Michelsohn</a>, <em><a class="existingWikiWord" href="/nlab/show/Spin+geometry">Spin geometry</a></em>, Princeton University Press (1989)</p> </li> <li id="AGP02"> <p>Marcelo Aguilar, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, Carlos Prieto, section 9 of <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href="http://tocs.ulb.tu-darmstadt.de/106999419.pdf">toc pdf</a>)</p> </li> <li id="Courtney04"> <p>Dennis Courtney, <em>A brief glance atK-theory</em>, 2004 (<a href="https://math.berkeley.edu/~hutching/teach/215b-2004/courtney.pdf">pdf</a>)</p> </li> <li id="Karoubi06"> <p><a class="existingWikiWord" href="/nlab/show/Max+Karoubi">Max Karoubi</a>, <em>K-theory. An elementary introduction</em>, lectures given at the Clay Mathematics Academy (<a href="https://arxiv.org/abs/math/0602082">arXiv:math/0602082</a>)</p> </li> <li id="Friedlander"> <p><a class="existingWikiWord" href="/nlab/show/Eric+Friedlander">Eric Friedlander</a>, <em>An introduction to K-theory</em> (emphasis on <a class="existingWikiWord" href="/nlab/show/algebraic+K-theory">algebraic K-theory</a>), 2007 (<a href="http://users.ictp.it/~pub_off/lectures/lns023/Friedlander/Friedlander.pdf">pdf</a>)</p> </li> <li id="Karpova09"> <p><a class="existingWikiWord" href="/nlab/show/Varvara+Karpova">Varvara Karpova</a>, <em>Complex topological K-theory</em>, 2009 (<a href="http://infoscience.epfl.ch/record/162450/files/karpova.semestre.hess2.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/KarpovaTopologicalKTheory.pdf" title="pdf">pdf</a>)</p> </li> <li id="Blair09"> <p>Chris Blair, <em>Some K-theory examples</em>, 2009 (<a href="http://www.maths.tcd.ie/~cblair/notes/kex.pdf">pdf</a>)</p> </li> <li id="Wirthmuller12"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Wirthm%C3%BCller">Klaus Wirthmüller</a>, <em>Vector bundles and K-theory</em>, 2012 (<a class="existingWikiWord" href="/nlab/files/wirthmueller-vector-bundles-and-k-theory.pdf" title="pdf">pdf</a>)</p> </li> <li> <p>Aderemi Kuku, <em>Introduction to K-theory and some applications</em> (<a href="https://www.math.ksu.edu/~zlin/kuku/Intro-Kthy.pdf">pdf</a>)</p> </li> </ul> <p>A textbook account of topological K-theory with an eye towards <a class="existingWikiWord" href="/nlab/show/operator+K-theory">operator K-theory</a> is section 1 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bruce+Blackadar">Bruce Blackadar</a>, <em><a class="existingWikiWord" href="/nlab/show/K-Theory+for+Operator+Algebras">K-Theory for Operator Algebras</a></em></li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/comparison+map+between+algebraic+and+topological+K-theory">comparison map between algebraic and topological K-theory</a> is discussed for instance in</p> <ul> <li id="Paluch"> <p>Michael Paluch, <em>Algebraic <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>-theory and topological spaces</em> K-theory 0471 (<a href="http://www.math.uiuc.edu/K-theory/0471/">web</a>)</p> </li> <li id="Rosenberg"> <p><a class="existingWikiWord" href="/nlab/show/Jonathan+Rosenberg">Jonathan Rosenberg</a>, <em>Comparison Between Algebraic and Topological K-Theory for Banach Algebras 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>, (<a href="http://www2.math.umd.edu/~jmr/algtopK.pdf">pdf</a>)</p> </li> </ul> <p>Discussion from the point of view of <a class="existingWikiWord" href="/nlab/show/smooth+stacks">smooth stacks</a> and <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> is in</p> <ul> <li id="BunkeNikolausVoelkl13"><a class="existingWikiWord" href="/nlab/show/Ulrich+Bunke">Ulrich Bunke</a>, <a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Michael+V%C3%B6lkl">Michael Völkl</a>, <em>Differential cohomology theories as sheaves of spectra</em>, Journal of Homotopy and Related Structures October 2014 (<a href="http://arxiv.org/abs/1311.3188">arXiv:1311.3188</a>)</li> </ul> <p>The proof of the <a class="existingWikiWord" href="/nlab/show/Hopf+invariant+one">Hopf invariant one</a> theorem in terms of topological K-theory is due to</p> <ul> <li id="AdamsAtiyah66"><a class="existingWikiWord" href="/nlab/show/Frank+Adams">Frank Adams</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>K-theory and the Hopf invariant</em>, Quart. J. Math. Oxford (2), 17 (1966), 31-38 (<a href="http://www.maths.ed.ac.uk/~aar/papers/adamatiy.pdf">pdf</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/Hopf+rings">Hopf rings</a> of <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> of topological K-theories:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>Bott periodicity and Hopf rings</em>, 1992 (<a href="https://neil-strickland.staff.shef.ac.uk/research/thesis.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/StricklandHopfRings.pdf" title="pdf">pdf</a>)</li> </ul> <h3 id="for_noncompact_spaces_2">For non-compact spaces</h3> <p>Topological K-theory of <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spaces">Eilenberg-MacLane spaces</a> is discussed in</p> <ul> <li id="AndersonHodgkin68"><a class="existingWikiWord" href="/nlab/show/Donald+W.+Anderson">Donald W. Anderson</a>, Luke Hodgkin <em>The K-theory of Eilenberg-Maclane complexes</em>, Topology, Volume 7, Issue 3, August 1968, Pages 317-329 (<a href="https://doi.org/10.1016/0040-9383(68">doi:10.1016/0040-9383(68)90009-8</a>90009-8))</li> </ul> <p>Topological topological K-theory of <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> is in</p> <ul> <li id="JackowskiOliver">Stefan Jackowski and Bob Oliver, <em>Vector bundles over classifying spaces of compact Lie groups</em> (<a href="http://hopf.math.purdue.edu/Jackowski-Oliver/bg-bu.pdf">pdf</a>)</li> </ul> <h3 id="dbrane_charge">D-brane charge</h3> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+D-brane+charge">K-theory classification of D-brane charge</a></em>.</p> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential</a> <a class="existingWikiWord" href="/nlab/show/topological+K-theory">K-theory</a> and its relation to <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> (see also <a href="D-brane#ReferencesKTheoryDescription">there</a>):</p> <ul> <li id="GradySati19a"><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Ramond-Ramond fields and twisted differential K-theory</em> (<a href="https://arxiv.org/abs/1903.08843">arXiv:1903.08843</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted</a> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential</a> <a class="existingWikiWord" href="/nlab/show/KO-theory">orthogonal</a> <a class="existingWikiWord" href="/nlab/show/topological+K-theory">K-theory</a> and its relation to <a class="existingWikiWord" href="/nlab/show/D-brane+charge">D-brane charge</a> in <a class="existingWikiWord" href="/nlab/show/type+I+string+theory">type I string theory</a> (on <a class="existingWikiWord" href="/nlab/show/orientifolds">orientifolds</a>):</p> <ul> <li id="GradySati19b"><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Twisted differential KO-theory</em> (<a href="https://arxiv.org/abs/1905.09085">arXiv:1905.09085</a>)</li> </ul> <h3 id="topological_phases_of_matter">Topological phases of matter</h3> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+topological+phases+of+matter">K-theory classification of topological phases of matter</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on November 16, 2024 at 10:22:07. 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