## The Relation of Cobordism to K-Theories |

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Page iii

To fit with quaternionic notation, the complex numbers are taken to

To fit with quaternionic notation, the complex numbers are taken to

**act**on the right and the inner product &, X is taken conjugate linear in the first variable and complex linear in the second. There is the graded exterior algebra /\ W ... Page v

(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)

(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

using (l.4), so that SU (n)

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

... H

... H

**acts**on the left hand side by l(2) q. Consider finally the case m = 4k + 2, n = 42 + 2. Define a left action of H on AW by q-Y = Y-q, so that we obtain a real vector space A V (3) A W. Here we write an element q as ex + A3 where ... Page 9

... and w E We we have > 0 and suppose (2.4) holds 2 2 _i \k+l _i \k % vo (X & Y) = (9 yox & y - (-1) *.x6 ; (x) + (-1) *.x & 9. X® (; )* - IIvlso II-II*,0.3%) (||v + w(I*)xG) Y. ~. The remark follows. Recall that U(n)

... and w E We we have > 0 and suppose (2.4) holds 2 2 _i \k+l _i \k % vo (X & Y) = (9 yox & y - (-1) *.x6 ; (x) + (-1) *.x & 9. X® (; )* - IIvlso II-II*,0.3%) (||v + w(I*)xG) Y. ~. The remark follows. Recall that U(n)

**acts**naturally on ...### 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