## The Relation of Cobordism to K-Theories |

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It is

It is

**proved**that given a compact (U, fr)-manifold *, there is a closed weakly almost complex manifold having the same Chern numbers 2n as M if and only if Td [*] is an integer; this makes use of recent theorems of Stong [23] and ... Page iv

... monomial X With X a X = (l-1) If X is a monomial then to X is the unique monomial X with This is readily seen from the definition of 'C'. (1.2) we have cox = (-1)***x for x s a v. Proof. It is sufficient to

... monomial X With X a X = (l-1) If X is a monomial then to X is the unique monomial X with This is readily seen from the definition of 'C'. (1.2) we have cox = (-1)***x for x s a v. Proof. It is sufficient to

**prove**(1.2) for monomials ... Page vi

The theorem is then

The theorem is then

**proved**. Let A°y F. x-A*ly, A*"w : x-A*w; similarly define R9°y and Rov. ... A. main purpose of this section is to**prove**the following. (2-1) THEOREM. There exist natural isomorphisms R(V + w) = R(w) ... Page x

... AW, and (2.l) is

... AW, and (2.l) is

**proved**. Return now to a single complex inner product space W of finite dimension. Given v e W there is F, : AW — AV defined by F,(X) = w a X. There is also its adjoint (F.)* : A V -> AV defined by <x, ...Page 24

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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