## The Relation of Cobordism to K-Theories |

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**similarly**for sl o and 0 °. On the other hand there are the Grothendieck—Atiyah-Hirzebruch periodic cohomology theories K"(. ), Ko" (-) of K-theory. The main point of Chapter I, then, is to define natural transformations + + a 0 (. Page

**Similarly**symplectic cobordism determines Ko"(...). The isomorphisms are generated by /*e, A. respectively. Warious topics are treated along the way, in particular cobordism characteristic classes. There is the sphere spectrum As. whose ... Page ii

**Similarly**given an SU(4k)—bundle there is constructed a Thom class t( 3 ) e Rö(M(s)), and given an SU(4k + 2)—bundle there is constructed a class s ( ; ) e KSp(M(s)). These Thom classes give rise to isomorphisms ~/ K(X) 2- K(M( £)) Ko ... Page vi

Let A°y F. x-A*ly, A*"w : x-A*w;

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 A*w and A'v. If n = 0 mod 4 then SU(n) acts on the real vector spaces Rody and R*Vy. 8. Page viii

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. If m = 4k and n = 4) # 2 then one sets up

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. If m = 4k and n = 4) # 2 then one sets up

**similarly**an isomorphism RW ...### 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