## The Relation of Cobordism to K-Theories |

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Page iii

The special unitary group SU(n) operates in a quaternionic

The special unitary group SU(n) operates in a quaternionic

**linear**fashion on AW in the first case, in a real**linear**fashion on RW in the second case. Fix, then, the complex inner product space W of dimension n. To fit with quaternionic ... Page iv

The map to is conjugate

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 SUstructure is c = el a ... a en. By a monomial of AW we mean an element ... Page v

We now identify U(n) with the group of

We now identify U(n) with the group of

**linear**maps g : V → V with <gu, gv) = <u, v × for all u, v e V. Then U (n) acts on AV by g(via © to o a vk) = gvi a . . . a gy. Identify the special unitary group SU(n) with the set of all g e U ... Page vi

using (l.4), so that SU (n) acts in a quaternionic

using (l.4), so that SU (n) acts in a quaternionic

**linear**fashion. If n = 4k, we have A* = 1. Hence AV = RW (3) R_(V). If X e RV, then 20,1) = -(/. X)1 = -Ki and Xi e R_(V). The theorem is then proved. Let A°y F. x-A*ly, A*"w : x-A*w; ...Page 10

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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