The Relation of Cobordism to K-TheoriesSpringer, 2006 M11 14 - 116 pages |
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Page 4
... given SU - structure . If n = 4k + 2 then AV becomes a right quaternionic vector space by defining Y⚫j = ( Y ) for Y & ɅV . Moreover SU ( n ) acts on AV in a quaternionic linear fashion . If n = 4k , let R ( V ) be all X & AV with μx ...
... given SU - structure . If n = 4k + 2 then AV becomes a right quaternionic vector space by defining Y⚫j = ( Y ) for Y & ɅV . Moreover SU ( n ) acts on AV in a quaternionic linear fashion . If n = 4k , let R ( V ) be all X & AV with μx ...
Page 5
... given SU - structures o identification A ( V + W ) = 1 2 AVAW of graded algebras , then V + W receives the SU - structure σ = σ ∞ σ According to section 1 , if 1 2 m = 2 mod 4 we consider AV as a Z - graded quaternionic vector space ...
... given SU - structures o identification A ( V + W ) = 1 2 AVAW of graded algebras , then V + W receives the SU - structure σ = σ ∞ σ According to section 1 , if 1 2 m = 2 mod 4 we consider AV as a Z - graded quaternionic vector space ...
Page 8
... Given v ɛ V there is F F F ( X ) V : AV → AV defined by = V ^ X. There is also its adjoint ( F ) * : ^ V → AV defined by < X , FY > = < F * X , Y > , all X , Y & Ʌv . Define 9 : AV → AV by 9 . F + ( F _ ) * . V V V V ( 2.3 ) Let V ...
... Given v ɛ V there is F F F ( X ) V : AV → AV defined by = V ^ X. There is also its adjoint ( F ) * : ^ V → AV defined by < X , FY > = < F * X , Y > , all X , Y & Ʌv . Define 9 : AV → AV by 9 . F + ( F _ ) * . V V V V ( 2.3 ) Let V ...
Page 9
... ) * • 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