## The Relation of Cobordism to K-Theories |

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Page ii

Formulas for the Chern character ch (J( ; ) are

Formulas for the Chern character ch (J( ; ) are

**obtained**. No claims for originality are made in this chapter; the methods have been well-known since the work of Atiyah-Hirzebruch [6], Dold [13], and others [7]. However since the results ... Page vi

In cases l and 4, the vector spaces and the isomorphisms are taken to be real linear, while in cases 2 and 3 they are taken to be quaternionic linear. For each v # 0 in V we also

In cases l and 4, the vector spaces and the isomorphisms are taken to be real linear, while in cases 2 and 3 they are taken to be quaternionic linear. For each v # 0 in V we also

**obtain**isomorphisms y, : A*wæ A*"v, m 9, Rodwo: R*"w, ... Page viii

Define a left action of H on AW by q-Y = Y-q, so that we

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; this is an anti-automorphism of H. There is a natural ...Page 12

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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