## The Relation of Cobordism to K-Theories |

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Page

... closed weakly almost complex

... closed weakly almost complex

**manifolds**M*, similarly for sl o and 0 °. On the other hand there are the Grothendieck—Atiyah-Hirzebruch periodic cohomology theories K"(. ), Ko" (-) of K-theory. The main point of Chapter I, then, ... Page

The elements or 0. fr are interpreted as bordism classes LM”) of compact (U, fr)—

The elements or 0. fr are interpreted as bordism classes LM”) of compact (U, fr)—

**manifolds**M”, where roughly a (U, fr)—**manifold**is a differentiable**manifold**M with a given complex structure on its stable tangent bundle to and a given ... Page

It is proved that given a compact (U, fr)-

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 i

U-

U-

**Manifolds**with Framed Boundaries . . . . . . . . . 69 The U-bordism groups no so & so o so to e o 'o e o 'o o o . . . . . . . . . . . . . 70 Characteristic numbers from K-theory . . . . . . . . . . . . . . 78 The theorem of Stong and ...Page 37

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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