## The Relation of Cobordism to K-Theories |

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... fr)—manifold is a differentiable manifold M with a given complex structure on its stable tangent bundle to and a given compatible framing of 27 restricted to the boundary O M. These bordism classes have Chern numbers and

... fr)—manifold is a differentiable manifold M with a given complex structure on its stable tangent bundle to and a given compatible framing of 27 restricted to the boundary O M. These bordism classes have Chern numbers and

**hence**a ... Page v

... gy > ,

... gy > ,

**hence**g T = C. g. 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 ... Page vi

**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; similarly define R9°y and Rov. If n = 2 mod 4 then SU(n) acts on the quaternionic vector spaces ... 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 identified. Page x

**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 be verified that "vow ) - = (F," & 1.### 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