The Relation of Cobordism to K-theoriesSpringer-Verlag, 1966 - 110 pages |
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Page 49
... n > 0 0 -1 -2 -3 n Sp -4 -5 -6 0 Z Z2 Z2 O Z Z Z2 Z2 We now turn to the problem of using ( 7.5 ) to give cobordism characteristic classes . We have first to understand MSp ( 1 ) . Let ૬ denote the Hopf Sp ( 1 ) -bundle over HP ( n ) ...
... n > 0 0 -1 -2 -3 n Sp -4 -5 -6 0 Z Z2 Z2 O Z Z Z2 Z2 We now turn to the problem of using ( 7.5 ) to give cobordism characteristic classes . We have first to understand MSp ( 1 ) . Let ૬ denote the Hopf Sp ( 1 ) -bundle over HP ( n ) ...
Page 51
... ( n + 1 ) ) 1 * sp ( HP ( n ) ) By the induction hypotheses , i * is an epimorphism , hence we get a commutative diagram • → Ã * Sp ( * Sp ( s4n + 4 ) * ( HP ( n + 1 ) ) 1 p ( HP ( n ) ) Sp i * Ω ( HP ( n ) ) → 0 Z Z JM Z 0 → H * ( s4n ...
... ( n + 1 ) ) 1 * sp ( HP ( n ) ) By the induction hypotheses , i * is an epimorphism , hence we get a commutative diagram • → Ã * Sp ( * Sp ( s4n + 4 ) * ( HP ( n + 1 ) ) 1 p ( HP ( n ) ) Sp i * Ω ( HP ( n ) ) → 0 Z Z JM Z 0 → H * ( s4n ...
Page 55
... ( HP ( n ) ) →→→ K0 * ( HP ( n ) ) Sp image of PA Pn maps the P of ( 8.1 ) into the element of ( 9.1 ) . ≈ Let p'n denote the 04 in Q Sp SU and it suffices in view of ( 5.2 ) to show that ( HP ( n ) ) → 241 , ( HP ( n ) ) . Then p ' ε n ...
... ( HP ( n ) ) →→→ K0 * ( HP ( n ) ) Sp image of PA Pn maps the P of ( 8.1 ) into the element of ( 9.1 ) . ≈ Let p'n denote the 04 in Q Sp SU and it suffices in view of ( 5.2 ) to show that ( HP ( n ) ) → 241 , ( HP ( n ) ) . Then p ' ε n ...
Contents
The Thom Isomorphism in Ktheory | 1 |
Cobordism Characteristic Classes | 38 |
UManifolds with Framed Boundaries | 69 |
Copyright | |
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abelian group algebra base point bordism bordism classes bordism groups bundle map ch+k characteristic classes Chern classes Chern numbers closed U-manifold cobordism cohomology theory complex inner product complex vector space composition consider COROLLARY CP(n CW pair X,A define denote diagram dimension Dold epimorphism fiber finite CW complex finite CW pair fr)-manifold framed manifold Hence homotopy class Hopf HP(n identified induced inner product space integer K-theory kernel KSP(X LEMMA line bundle linear M²n map f Milnor monomial MSU 4k MU(k multiplicative cohomology theory n)-bundle ñ¹ 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 vector space bundle Z-graded εΩ Ωυ