## The Relation of Cobordism to K-theories |

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Page 48

Characteristic

Characteristic

**classes**in cobordisn . In this section we set up central tools for this chapter . Recall that in section 5 we have considered the cohomology theories ( :) SU 4k k of symplectic , special unitary , unitary cobordism .Page 69

In section 12 we consider the bordism group 2 of closed U - manifolds of dimension n ; the elements of nu are the bordism

In section 12 we consider the bordism group 2 of closed U - manifolds of dimension n ; the elements of nu are the bordism

**classes**[ M ] of closed differentiable manifolds n with a given complex structure on the stable tangent bundle .Page 75

As with ordinary Chern

As with ordinary Chern

**classes**, there is the formula 2 n Swl 5 + 7 ) = £ s ( 5 ) • scroly ) w ' + w !! If mh is a U - manifold we may consider the stable tangent bundle T + ( 2k k ( 2k - n ) as a complex vector space bundle .### What people are saying - Write a review

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### Contents

CONTENTS 1 | 1 |

Tensor products of exterior algebras | 5 |

Application to bundles | 11 |

Copyright | |

16 other sections not shown

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### Common terms and phrases

abelian According acts Adams assigning associated base point basis bordism bordism classes BSp(n Chapter characteristic Chern classes Chern numbers classes closed U-manifold cobordism coefficient commutativity compact U,fr)-manifold complex vector space composition consider construction COROLLARY CP(n define definition denote diagram dimension element equivalence exact exists fact fiber finite CW complex follows function give given Hence holds homo topy homomorphism HP(n identified induced integer isomorphism K-theory KSP(X line bundle linear manifold Moreover MU(K Namely natural Note obtain particular partition polynomial Proof prove quaternionic represented respectively ring seen Similarly stable tangent bundle stably framed structure su(n sufficient Suppose theorem Thom Todd genus trivial U-structure U,fr U(n)-bundle unique universal vector space bundle