The Relation of Cobordism to K-TheoriesSpringer, 2006 M11 14 - 116 pages |
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Page 3
... Proof . It is sufficient to prove ( 1.2 ) for monomials . monomial , TX is the unique monomial with XA TX = σ . Tx ^ x = ( -1 ) k ( n - k ) σ and Define an operator μ : ^ kv conjugate linear . 2x ( −1 ) k ( n − k ) x . For X a → ^ n ...
... Proof . It is sufficient to prove ( 1.2 ) for monomials . monomial , TX is the unique monomial with XA TX = σ . Tx ^ x = ( -1 ) k ( n - k ) σ and Define an operator μ : ^ kv conjugate linear . 2x ( −1 ) k ( n − k ) x . For X a → ^ n ...
Page 4
... Proof . __ = Tg and Mb = вре g · ^ If g ε SU ( n ) , then g7 = ɛ MB From < X , Y > = ( o , XA Y > we get < g TX , gY > = = < o , gX ^ gY > hence g = Tg . = < 7gX , gy > , It follows immediately that gμ = μ8 . ( 1.5 ) THEOREM . Consider ...
... Proof . __ = Tg and Mb = вре g · ^ If g ε SU ( n ) , then g7 = ɛ MB From < X , Y > = ( o , XA Y > we get < g TX , gY > = = < o , gX ^ gY > hence g = Tg . = < 7gX , gy > , It follows immediately that gμ = μ8 . ( 1.5 ) THEOREM . Consider ...
Page 6
... proof of ( 2.1 ) is based on the following lemma . r S ( 2.2 ) LEMMA . If X & AV and Y ε W then μ ( X ® X ) ( -1 ) ms = 1 M = ( X ) R μ2 ( X ) 2 where μ , M1 , M2 denote the maps of section 1 for ^ ( V + W ) = ^ ▽ ® ^ W , AV , AW ...
... proof of ( 2.1 ) is based on the following lemma . r S ( 2.2 ) LEMMA . If X & AV and Y ε W then μ ( X ® X ) ( -1 ) ms = 1 M = ( X ) R μ2 ( X ) 2 where μ , M1 , M2 denote the maps of section 1 for ^ ( V + W ) = ^ ▽ ® ^ W , AV , AW ...
Page 8
... Proof . The element vw corresponds to v 1 + 1 w ε ^ V ^ W . where f : Fy + w ( X® Y ) = ( ▽ ^ 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 ...
... Proof . The element vw corresponds to v 1 + 1 w ε ^ V ^ W . where f : Fy + w ( X® Y ) = ( ▽ ^ 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 ...
Page 9
... Proof . As an exercise the reader may check this in case dim V = 1. If dim V > 1 split V as the direct sum of orthogonal sub- space V1 + V2 V2 where dim V1 > 0 , dim V2 > 0 and suppose ( 2.4 ) holds for V , and V2 * 1 ( 9 ) 2 ( x® ¥ ) V ...
... Proof . As an exercise the reader may check this in case dim V = 1. If dim V > 1 split V as the direct sum of orthogonal sub- space V1 + V2 V2 where dim V1 > 0 , dim V2 > 0 and suppose ( 2.4 ) holds for V , and V2 * 1 ( 9 ) 2 ( x® ¥ ) V ...
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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