## The Relation of Cobordism to K-Theories |

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Page v

**Consider**the complex inner product space W of dimension n, with given SU-structure a e A*w. If n = 4k + 2 then AW becomes a right quaternionic vector space by defining Y j = A (Y) for Y e AV. Moreover SU(n) acts on AV in a ... Page vi

... mod 4 we

... mod 4 we

**consider**AW as **** quaternionic vector space while if m = 0 mod 4 we obtain a Ze-graded real vector space RW. A. main purpose of this section is to prove the following. (2-1) THEOREM. There exist natural isomorphisms R(V + ... Page vii

Note that if m is even then A = /*1 (3) A 2. We

Note that if m is even then A = /*1 (3) A 2. We

**consider**now the proof of (2.l.) for m = 4k and n = 41. There is a natural homomorphism * : A W © Aw -> AW ... Page viii

**Consider**finally the case m = 4k + 2, n = 42 + 2. Define a left action of H on AW by q-Y = Y-q, so that we obtain a real vector space A V (3) A W. Here we write an element q as ex + A3 where of , /*e C and aerine q = 2^ –73 j; ... Page x

... X. There is also its adjoint (F.)* : A V -> AV defined by <x, KYX = < Fox,Y > , all x,Y = Av. Define : W —2. W b = F + (F )+. %, A A y °, V (F, (2-3) Let V and W be complex inner product spaces, let v e V, w e W and

... X. There is also its adjoint (F.)* : A V -> AV defined by <x, KYX = < Fox,Y > , all x,Y = Av. Define : W —2. W b = F + (F )+. %, A A y °, V (F, (2-3) Let V and W be complex inner product spaces, let v e V, w e W and

**consider**v * W e ...### What people are saying - Write a review

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### Contents

ii | |

vi | |

Application to bundles | 11 |

Thom classes of line bundles | 18 |

Cobordism and homomorphisms into Ktheory | 25 |

The homomorphism e | 30 |

Cobordism Characteristic Classes 58 | 38 |

A theorem of Dold | 39 |

UManifolds with Framed Boundaries | 69 |

The Ubordism groups nº ſº ſº º ſº tº e º º e º º º º | 70 |

Characteristic numbers from Ktheory | 78 |

The theorem of Stong and Hattori | 82 |

Umanifolds with stably framed boundaries | 91 |

The bordism groups Q V fr | 96 |

The groups nº | 105 |

in a | 108 |

Characteristic classes in cobordism | 48 |

Characteristic classes in Ktheory | 52 |

A cobordism interpretation for K X | 65 |

Bibliography | 111 |

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### Common terms and phrases

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