The Relation of Cobordism to K-theoriesSpringer-Verlag, 1966 - 110 pages |
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Page 51
... hence 1 * ( Pn + 1 ) n + 1 = 0 by ( 7.1 ) . with 77 * ) = ( Pp + 1 ) n + 1 . ) n + l . Then Then It may be seen by induction on with generator any element ã * Hence ( s4n + 4 ) has basis Sp * diagram that ( HP ( n + 1 ) ) Sp ( HP ( n 1 ...
... hence 1 * ( Pn + 1 ) n + 1 = 0 by ( 7.1 ) . with 77 * ) = ( Pp + 1 ) n + 1 . ) n + l . Then Then It may be seen by induction on with generator any element ã * Hence ( s4n + 4 ) has basis Sp * diagram that ( HP ( n + 1 ) ) Sp ( HP ( n 1 ...
Page 66
... hence U + has Todd genus T [ M ] = 1 8n 8n 8n εΩ 8n 1 according to ( 6.5 ) . It is then sufficient to switch to bordism . SU Denote by the bordism ring of closed SU - manifolds ( denoted by in [ 12 ] ) . We 8n must prove that if [ Mon ) ...
... hence U + has Todd genus T [ M ] = 1 8n 8n 8n εΩ 8n 1 according to ( 6.5 ) . It is then sufficient to switch to bordism . SU Denote by the bordism ring of closed SU - manifolds ( denoted by in [ 12 ] ) . We 8n must prove that if [ Mon ) ...
Page 104
... hence Hence Ta [ м8k + 4 ] is even . Since SU SU = 0 and 8k + 3 8k + 5 = 0 [ 12 ] , we have the diagram Ω SU 8k + 4 u SU , fr 8k + 4 → nf Ofr 8k + 3 ( 1/2 ) Td Q and we see from ( 16.4 ) that ( 1/2 ) Td maps image u into Z. get an ...
... hence Hence Ta [ м8k + 4 ] is even . Since SU SU = 0 and 8k + 3 8k + 5 = 0 [ 12 ] , we have the diagram Ω SU 8k + 4 u SU , fr 8k + 4 → nf Ofr 8k + 3 ( 1/2 ) Td Q and we see from ( 16.4 ) that ( 1/2 ) Td maps image u into Z. get an ...
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