The Relation of Cobordism to K-TheoriesSpringer, 2006 M11 14 - 116 pages |
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Page 5
... SU ( n ) acts in a quaternionic linear fashion . 2 If = 1. Hence AV = RVR_ ( V ) . If X & RV , then μ ( xi ) = - ( μ ... n = 2 mod 4 then SU ( n ) od ет spaces V and ^ v . 2k V ; acts on the quaternionic vector If n = 0 mod 4 then SU ( n ) ...
... SU ( n ) acts in a quaternionic linear fashion . 2 If = 1. Hence AV = RVR_ ( V ) . If X & RV , then μ ( xi ) = - ( μ ... n = 2 mod 4 then SU ( n ) od ет spaces V and ^ v . 2k V ; acts on the quaternionic vector If n = 0 mod 4 then SU ( n ) ...
Page 7
... SU ( n ) on the two sides are identified . RV If m = 4k and n = 47 + 2 then one sets up similarly an isomorphism R AW ≈ ( V + W ) of quaternionic vector spaces , where q & H acts on the left hand side by 10 q . Consider finally the ...
... SU ( n ) on the two sides are identified . RV If m = 4k and n = 47 + 2 then one sets up similarly an isomorphism R AW ≈ ( V + W ) of quaternionic vector spaces , where q & H acts on the left hand side by 10 q . Consider finally the ...
Page 9
... ( 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 or ... SU - structure given by σ & ^ ε Av ; there V Ev8 . is the induced operator : ^ V → ^ V . = V ( −1 ) * ( F ̧ ) * 9 ...
... ( 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 or ... SU - structure given by σ & ^ ε Av ; there V Ev8 . is the induced operator : ^ V → ^ V . = V ( −1 ) * ( F ̧ ) * 9 ...
Page 10
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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