The Relation of Cobordism to K-TheoriesSpringer, 2006 M11 14 - 116 pages |
From inside the book
Results 1-5 of 41
Page 2
... follows : fix X & Ʌy and let Y vary overn - ky so that < , XY ) is a linear map TX to be the unique element of n - ky such that n - ky VC ; define < ~ X , Y ) = ( 0 , x ^ Y > , all Y ɛ ^ n - kv . Ʌn - ky . It is then seen that the above ...
... follows : fix X & Ʌy and let Y vary overn - ky so that < , XY ) is a linear map TX to be the unique element of n - ky such that n - ky VC ; define < ~ X , Y ) = ( 0 , x ^ Y > , all Y ɛ ^ n - kv . Ʌn - ky . It is then seen that the above ...
Page 3
... 1 ) * x where X VAIN r = k ( k 1 ) / 2 + ( n k ) ( n - k - 1 ) / 2 + k ( n → k ) = k ( n - 1 ) / 2 + ( n- k ) ( n - 1 ) / 2 = n ( n - 1 ) / 2 . The remark follows . We now identify U ( n ) with the group 3 מא Application to bundles.
... 1 ) * x where X VAIN r = k ( k 1 ) / 2 + ( n k ) ( n - k - 1 ) / 2 + k ( n → k ) = k ( n - 1 ) / 2 + ( n- k ) ( n - 1 ) / 2 = n ( n - 1 ) / 2 . The remark follows . We now identify U ( n ) with the group 3 מא Application to bundles.
Page 4
... follows from ( 1.3 ) that = -1 . Also is conjugate linear so that Xij = = μ ( xi ) ( x ) = -Xji . It follows that there is defined an action of the quaternions H on AV , and AV is a quaternionic vector space . Consider g ɛ SU ( n ) ...
... follows from ( 1.3 ) that = -1 . Also is conjugate linear so that Xij = = μ ( xi ) ( x ) = -Xji . It follows that there is defined an action of the quaternions H on AV , and AV is a quaternionic vector space . Consider g ɛ SU ( n ) ...
Page 6
... follows . and Y is 1 2 Note that if m is even then μ = μ1 R μ 2 ° риз We consider now the proof of ( 2.1 ) for m = 4k and n = 41 . There is a natural homomorphism Y : AV 1W → AV R ^ W γ : ο AW R - C whose kernel is generated by all Xi ...
... follows . and Y is 1 2 Note that if m is even then μ = μ1 R μ 2 ° риз We consider now the proof of ( 2.1 ) for m = 4k and n = 41 . There is a natural homomorphism Y : AV 1W → AV R ^ W γ : ο AW R - C whose kernel is generated by all Xi ...
Page 9
... follows . p * . 2 ( 2.4 ) For each v ɛ V we have ( 9 ) 2 = || v || 2 1 . 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 ...
... follows . p * . 2 ( 2.4 ) For each v ɛ V we have ( 9 ) 2 = || v || 2 1 . 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 ...
Other editions - View all
Common terms and phrases
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