## The Relation of Cobordism to K-Theories |

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The main point of Chapter I, then, is to define natural transformations + + a 0 (...) — Ko" (.) A. : o () → *t.) of cohomology theories. Such transformations have been folk

The main point of Chapter I, then, is to define natural transformations + + a 0 (...) — Ko" (.) A. : o () → *t.) of cohomology theories. Such transformations have been folk

**theorems**since the work of Atiyah-Hirzebruch [6], Dold [lo), ... Page

It is proved that given a compact (U, fr)-manifold *, there is a closed weakly almost complex manifold having the same Chern numbers 2n as M if and only if Td [*] is an integer; this makes use of recent

It is proved that given a compact (U, fr)-manifold *, there is a closed weakly almost complex manifold having the same Chern numbers 2n as M if and only if Td [*] is an integer; this makes use of recent

**theorems**of Stong [23] and ... Page i

U-Manifolds with Framed Boundaries . . . . . . . . . 69 The U-bordism groups no so & so o so to e o 'o e o 'o o o . . . . . . . . . . . . . 70 Characteristic numbers from K-theory . . . . . . . . . . . . . . 78 The

U-Manifolds with Framed Boundaries . . . . . . . . . 69 The U-bordism groups no so & so o so to e o 'o e o 'o o o . . . . . . . . . . . . . 70 Characteristic numbers from K-theory . . . . . . . . . . . . . . 78 The

**theorem**of Stong and ... Page v

(1.5)

(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 Y e AV. Moreover SU(n) acts on AV in ... Page vi

The

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