## The Relation of Cobordism to K-Theories |

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Page iii

... fix X e A*w and let Y vary over A **w so that 3-,x a Y > is a linear map An-ky — C; define ^ X to be the unique element of A*w such that —k Kox,Y > = <g-,x a y > , all Y e A**v. It is then

... fix X e A*w and let Y vary over A **w so that 3-,x a Y > is a linear map An-ky — C; define ^ X to be the unique element of A*w such that —k Kox,Y > = <g-,x a y > , all Y e A**v. It is then

**seen**that the above equation holds for. Page iv

It is then

It is then

**seen**that the above equation holds for all Y e /NW. The map to is conjugate linear. For 43 (Xa), Y > = a Ko–,X a Y > = <! & X)a, Y > and 7 (Xa) = (~ X)a. Fix an orthonormal basis el, ''', en of W such that the given ... Page viii

It is

It is

**seen**that if y e Kernel 5 , then (169 i)x = -(1 & 1)y. If also y e R(W) (X) R(W) then the left hand side belongs to R R_(W) & R(W) and the right hand side to R(W) ©s R_(W) by (l.5). Hence y = 0, and RW ($) RW maps monomorphically ... Page x

/*(X (96 y) = A1x &c. Ae y = x3 & X = -(x & Y). But -(Xj ©o jY) = X & Y. It is thus

/*(X (96 y) = A1x &c. Ae y = x3 & X = -(x & Y). But -(Xj ©o jY) = X & Y. It is thus

**seen**that Kernel 3 ->R-(V + W). A check of dimensions reveals that we have Kernel 3 = R_(W + W.), Since of f is an epimorphism.Page 13

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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