## The Relation of Cobordism to K-Theories |

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In Chapter I we treat the Thom

In Chapter I we treat the Thom

**isomorphism**in K-theory. The families U, SU, Sp of unitary, special unitary, symplectic groups generate spectra MU, MSU, MSp of Thom spaces. In the fashion of G. W. Whitehead [26 J, each spectrum generates ... Page

The

The

**isomorphisms**are generated by /*e, A. respectively. Warious topics are treated along the way, in particular cobordism characteristic classes. There is the sphere spectrum As. whose homology groups are the framed bordism groups ... Page

However the recent work of AndersonBrown-Peterson is a notable example of the application of K-theory to CONTENTS Chapter I. The Thom

However the recent work of AndersonBrown-Peterson is a notable example of the application of K-theory to CONTENTS Chapter I. The Thom

**Isomorphism**in K-theory. Page i

CONTENTS Chapter I. The Thom

CONTENTS Chapter I. The Thom

**Isomorphism**in K-theory. * * * * * * * * * * * * * * 1 1. 2. Exterior Algebra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Tensor products of exterior algebras . Page ii

CHAPTER I. THE THOM

CHAPTER I. THE THOM

**ISOMORPHISM**IN K–THEORY. Given a U(n)-bundle 5 over a finite CW complex X there is ... These Thom classes give rise to**isomorphisms**~/ K(X) 2- K(M( £)) Ko (x) = Ko (M(s)) aKO (X) o KSp(M(s)) in the three cases.### 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