## The Relation of Cobordism to K-Theories |

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Page ii

Given a U(

Given a U(

**n**)-bundle 5 over a finite CW complex X there is conStructed an element J ( ; ) 8. Kūstā )) where M( ; ) is the Thom space of 5 ; we call J ( 3 ) the Thom class of 5 . Similarly given an**SU**(4k)—bundle there is constructed a ... Page iii

n = 4k then a real vector subspace RV of /\ W is selected so that /\W is identified with the complexification of RV. The special unitary group

n = 4k then a real vector subspace RV of /\ W is selected so that /\W is identified with the complexification of RV. The special unitary group

**SU**(**n**) operates in a quaternionic linear fashion on AW in the first case, in a real linear ... Page v

... v × for all u, v e V. Then U (n) acts on AV by g(via © to o a vk) = gvi a . . . a gy. Identify the special unitary group

... v × for all u, v e V. Then U (n) acts on AV by g(via © to o a vk) = gvi a . . . a gy. 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 = g/. Page vi

using (l.4), so that

using (l.4), so that

**SU**(**n**) acts in a quaternionic linear fashion. If n = 4k, we have A* = 1. 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; ... 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) ×

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.### 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