## The Relation of Cobordism to K-Theories |

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### Contents

ii | |

Tensor products of exterior algebras | 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 group algebra base point bordism classes bordism groups bundle map characteristic classes Chern classes Chern numbers closed U-manifold cobordism cobordism theories coefficient group commutativity holds complex inner product complex vector space composition consider COROLLARY CP(n define denote diagram dimension epimorphism exterior algebra fiber finite CW complex finite CW pair follows readily fr)—manifold framed manifold Hence Hirzebruch homotopy class Hopf HP(n identified induced inner product space integer isomorphism 9 K-theory kernel KSp(X LEMMA line bundle Mºn MU(k MU(n multiplicative cohomology theory natural map partition polynomial Proof quaternionic vector space ring homomorphism Similarly sºn Sp(l Sp(m)—bundle stable tangent bundle stably framed manifold SU(n Suppose symmetric symplectic Thom class Thom isomorphism Thom space Todd genus trivial U-structure U(n)-bundle unitary vector space bundle wºn Ze-graded