## The Relation of Cobordism to K-Theories |

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The main point of Chapter I, then, is to

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 iii

**DEFINITION**. 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 ... Page iv

CX a X = (–1) o and Tox = (–1)

CX a X = (–1) o and Tox = (–1)

**Define**an operator A : A*w – An-ow by A = a x. Then/ is conjugate linear. n(n-1)/2 (l. 3) We have Adox = (–1) X for X e /\V. Proof. It is seen from (l. 2) that A*x -: (-1)*x where r - k(k - 1)/2 + (n - k) ... Page v

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

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 vi

Let A°y F. x-A*ly, A*"w : x-A*w; similarly

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.### 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