The Relation of Cobordism to K-TheoriesSpringer, 2006 M11 14 - 116 pages |
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... U , fr n are interpreted as bordism classes [ M ] of compact ( U , fr ) -manifolds M , where roughly a ( U , fr ) -manifold is a differentiable manifold M with a given complex structure on its stable tangent bundle T and a given ...
... U , fr n are interpreted as bordism classes [ M ] of compact ( U , fr ) -manifolds M , where roughly a ( U , fr ) -manifold is a differentiable manifold M with a given complex structure on its stable tangent bundle T and a given ...
Page 3
... structure is σ = e1 Λ ... element X = - erl Λ · A erk where г1 < ... < Ik if X and Y are monomials , then 1 if Y X ... U ( n ) with the group 3 מא Application to bundles.
... structure is σ = e1 Λ ... element X = - erl Λ · A erk where г1 < ... < Ik if X and Y are monomials , then 1 if Y X ... U ( n ) with the group 3 מא Application to bundles.
Page 4
P. E. Conner, E. E. Floyd. We now identify U ( n ) with the group of linear maps g : V → V with < gu , gv ) 8 ( 51 8 ( 1 ^ • • • ^ k ) Κ < u ... structure . If n = 4k + 2 then AV becomes a right quaternionic vector space by defining Y⚫j = ( ...
P. E. Conner, E. E. Floyd. We now identify U ( n ) with the group of linear maps g : V → V with < gu , gv ) 8 ( 51 8 ( 1 ^ • • • ^ k ) Κ < u ... structure . If n = 4k + 2 then AV becomes a right quaternionic vector space by defining Y⚫j = ( ...
Page 9
... U ( n ) acts naturally on V. ( 2.5 ) Proof . For any v ɛ V and g ɛ U ( n ) we have 9 . Since F ( X ) = V ^ X we have ... structure given by σ & ^ ε Av ; there V Ev8 . is the induced operator : ^ V → ^ V . = V ( −1 ) * ( F ̧ ) * 9 ...
... U ( n ) acts naturally on V. ( 2.5 ) Proof . For any v ɛ V and g ɛ U ( n ) we have 9 . Since F ( X ) = V ^ X we have ... structure given by σ & ^ ε Av ; there V Ev8 . is the induced operator : ^ V → ^ V . = V ( −1 ) * ( F ̧ ) * 9 ...
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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