## The Relation of Cobordism to K-Theories |

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Page iv

... Moreover given a monomial X there is a unique monomial X With X a X = (l-1) If X is a monomial then to X is the unique monomial X with This is readily seen from the definition of 'C'. (1.2) we have cox = (-1)***x for x s a v.

... Moreover given a monomial X there is a unique monomial X With X a X = (l-1) If X is a monomial then to X is the unique monomial X with This is readily seen from the definition of 'C'. (1.2) we have cox = (-1)***x for x s a v.

**Proof**. Page v

**Proof**. Consider the case n = 4k + 2. It follows from (l. 3) that A* = -l. Also A is conjugate linear so that Xij = A-(xi) = -(/ex) i = -Xji. It follows that there is defined an action of the quaternions H on AW, and AW is a quaternionic ... Page vii

4k + 2 The

4k + 2 The

**proof**of (2.1) is based on the following lemma. r S (2.2) LEMMA. If X e A V and Y e AW then A G. Gy) = (-1)" /,000 A, x) where A , A-1, /* 2 denote the maps of section l for A (W 4 W) = AV (9/\W, AV, AW respectively.**Proof**. Page x

**Proof**. The element v + w corresponds to v (3) l + 1 (3) w e AWG) AW. Hence F.G. & Y) = (v a x) & y + (-1)*x (w a Y) = F,(x) & y + (-1)*x & F.C.), *vow F F, Gol + Áe (1 (3) F.) where É, Av Go Aw -> Avo aw maps x& Y into (-1)*xó Y. It may ... Page 9

**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, X 0, dim W l 2 l 2 for Vl and We' For V & Vl and w E We we have > 0 and suppose ...### What people are saying - Write a review

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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