## The Relation of Cobordism to K-Theories |

### From inside the book

Results 1-5 of 41

Page iii

By an SU-structure for W we shall mean a unit vector O E A*w; suppose an SU-structure has been fixed for W. Define a real linear map C': A*w -> An-ky as

By an SU-structure for W we shall mean a unit vector O E A*w; suppose an SU-structure has been fixed for W. Define a real linear map C': A*w -> An-ky as

**follows**: fix X e A*w and let Y vary over A **w so that 3-,x a Y > is a linear map ... Page iv

It is seen from (l. 2) that A*x -: (-1)*x where r - k(k - 1)/2 + (n - k) (n - k - 1)/2 + k(n - k) k(n - 1)/2 + (n - k) (n − 1)/2 = n(n - l)/2. The remark

It is seen from (l. 2) that A*x -: (-1)*x where r - k(k - 1)/2 + (n - k) (n - k - 1)/2 + k(n - k) k(n - 1)/2 + (n - k) (n − 1)/2 = n(n - l)/2. The remark

**follows**. We now identify U(n) with the group of linear maps. Page v

It

It

**follows**immediately that g/ = A g. (1.5) THEOREM. Consider the complex inner product space W of dimension n, with given SU-structure a e A*w. If n = 4k + 2 then AW becomes a right quaternionic vector space by defining Y j = A (Y) for ... Page vii

... o: and the result

... o: and the result

**follows**. Note that if m is even then A = /*1 (3) A 2. We consider now the proof of (2.l.) for m = 4k and n = 41. There is a natural homomorphism * : A W © Aw -> AW ... Page 9

... Since A” - A . The remark

... Since A” - A . The remark

**follows**. (2.a) For each v e v we have (; )* = ||v|| 1. Proof. As an exercise the reader may check this in case dim W = l. If dim W X 1 split W as the direct sum of orthogonal subspace W., + W., where dim W, ...### What people are saying - Write a review

We haven't found any reviews in the usual places.

### 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 |

### Other editions - View all

### 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