## The Relation of Cobordism to K-Theories |

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should be noted that on the coefficient groups, 0 ° – K* (pt) = z /*e it u (pt) = is

should be noted that on the coefficient groups, 0 ° – K* (pt) = z /*e it u (pt) = is

**identified**up to sign with the Todd genus Td : O." -> Z. In Chapter II we show among other things that to: cobordism theories determine the K-theories. Page iii

n = 4k then a real vector subspace RV of /\ W is selected so that /\W is

n = 4k then a real vector subspace RV of /\ W is selected so that /\W is

**identified**with the complexification of RV. The special unitary group SU(n) operates in a quaternionic linear fashion on AW in the first case, in a real linear ... Page vi

Let W and W be complex inner product spaces of dimension m, n respectively, with given SU-structures a-, and a- . Using the

Let W and W be complex inner product spaces of dimension m, n respectively, with given SU-structures a-, and a- . Using the

**identification**A (V + W) = /\V 6) AW of graded asso, then W + W. receives the SU-structure os = o G9 o' . Page viii

Hence y = 0, and RW ($) RW maps monomorphically into R(W + W). Since the two are seen to have the same dimension, then RW ©s RW 33 R(W + W). It is also seen that the actions of SU(m) × SU(n) on the two sides are

Hence y = 0, and RW ($) RW maps monomorphically into R(W + W). Since the two are seen to have the same dimension, then RW ©s RW 33 R(W + W). It is also seen that the actions of SU(m) × SU(n) on the two sides are

**identified**. Page x

... A V -> AV defined by <x, KYX = < Fox,Y > , all x,Y = Av. Define : W —2. W b = F + (F )+. %, A A y °, V (F, (2-3) Let V and W be complex inner product spaces, let v e V, w e W and consider v * W e V + W. Using the

... A V -> AV defined by <x, KYX = < Fox,Y > , all x,Y = Av. Define : W —2. W b = F + (F )+. %, A A y °, V (F, (2-3) Let V and W be complex inner product spaces, let v e V, w e W and consider v * W e V + W. Using the

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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