## The Relation of Cobordism to K-Theories |

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should be noted that on the coefficient groups, 0 ° – K* (pt) = z /*e it

should be noted that on the coefficient groups, 0 ° – K* (pt) = z /*e it

**u**(pt) = is identified up to sign with the Todd ... where roughly a (**U**, fr)—manifold is a differentiable manifold M with a given complex**structure**on its stable ... Page v

Identify the special unitary group SU(n) with the set of all g e

Identify the special unitary group SU(n) with the set of all g e

**U**(n) for which g (o-) = GT . (1.4) If g e SU(n), then god = ? g and A g ... Consider the complex inner product space W of dimension n, with given SU-**structure**a e A*w. Page 9

Recall that

Recall that

**U**(n) acts naturally on W. 2. F d : or e o (2.5) For any v e V and g e**U**(n) we have $ gy * * g ?, Proof. ... Hence go y = } gvē" Suppose now that W has an SU-**structure**given by or e A”v; there is the induced operator / ...Page 11

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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