The Relation of Cobordism to K-TheoriesSpringer, 2006 M11 14 - 116 pages |
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... characteristic classes . There is the sphere spectrum , whose homology groups are the fr framed bordism groups Ω * ( • ) . Q fr n ( point ) The spectrum - The coefficient group Ω fr are just the stable stems π7 ( s * ) , k large . n n + ...
... characteristic classes . There is the sphere spectrum , whose homology groups are the fr framed bordism groups Ω * ( • ) . Q fr n ( point ) The spectrum - The coefficient group Ω fr are just the stable stems π7 ( s * ) , k large . n n + ...
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... Characteristic classes in cobordism 9. Characteristic classes in K - theory 10. A cobordism interpretation for K * ( X ) 1 1 5 11 18 • 25 • 30 38 39 48 52 59 11. Mappings into spheres 65 Chapter III . U - Manifolds with Framed ...
... Characteristic classes in cobordism 9. Characteristic classes in K - theory 10. A cobordism interpretation for K * ( X ) 1 1 5 11 18 • 25 • 30 38 39 48 52 59 11. Mappings into spheres 65 Chapter III . U - Manifolds with Framed ...
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... There is a natural epimorphism AV d ' : ^ V R ^ w → Av R ® C 1W . H If X ε AV and Y ɛ Aw , then ɣ ' maps μ ( X ® Y ) and X Y into the same value . For we have ~ ( x R Y ) = Mix ® c 7 Cobordism Characteristic Classes A theorem of Dold.
... There is a natural epimorphism AV d ' : ^ V R ^ w → Av R ® C 1W . H If X ε AV and Y ɛ Aw , then ɣ ' maps μ ( X ® Y ) and X Y into the same value . For we have ~ ( x R Y ) = Mix ® c 7 Cobordism Characteristic Classes A theorem of Dold.
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... ) = ( ▽ ^ x ) © Y + ( -1 ) * x ( WAY ) = F ( X ) Y + V ( −1 ) * x ® F ̧ ( Y ) , Fv + W = F ß 1 + ẞo ( 1F ) V \ V® AW → Av® AW maps X Y into ( -1 ) * x® Y. It may be verified that ( F = V + W ß 8 Characteristic classes in cobordism.
... ) = ( ▽ ^ x ) © Y + ( -1 ) * x ( WAY ) = F ( X ) Y + V ( −1 ) * x ® F ̧ ( Y ) , Fv + W = F ß 1 + ẞo ( 1F ) V \ V® AW → Av® AW maps X Y into ( -1 ) * x® Y. It may be verified that ( F = V + W ß 8 Characteristic classes in cobordism.
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... ) * • 8 • Hence g9 = 9 gvg . Suppose now that V has an SU - structure given by σ & ^ ε Av ; there V Ev8 . is the induced operator : ^ V → ^ V . = V ( −1 ) * ( F ̧ ) * 9 Characteristic classes in K-theory A cobordism interpretation for K*
... ) * • 8 • Hence g9 = 9 gvg . Suppose now that V has an SU - structure given by σ & ^ ε Av ; there V Ev8 . is the induced operator : ^ V → ^ V . = V ( −1 ) * ( F ̧ ) * 9 Characteristic classes in K-theory A cobordism interpretation for K*
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abelian group algebra base point bordism bordism classes BSp(n bundle map c₁ ch+k characteristic classes Chern classes Chern numbers closed U-manifold cobordism coefficient groups cohomology theory complex inner product complex vector space consider COROLLARY CP(n CW pair X,A define denote diagram dimension Dold element epimorphism ɛ K(M finite CW complex finite CW pair framed manifold Hence Hirzebruch homotopy class Hopf HP(n identified induced inner product space integer K-theory kernel KSP(X LEMMA line bundle M²n map f module monomial MSU 4k MU(k multiplicative cohomology theory n+2k ñ¹(x P₁ partition polynomial Proof quaternionic quaternionic vector space Similarly stable tangent bundle stably framed manifold SU(n Suppose theorem Thom class Thom space Todd genus trivial U-structure U(n)-bundle unitary vector space bundle Z-graded εΩ Ωυ हु