## The Relation of Cobordism to K-Theories |

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The cohomology theories are

The cohomology theories are

**denoted**by 0 (-), a.o.), a. ( . ) and are called cobordism theories; the homology theories are aenoa by do.), no"(...), o:*C.) and are called bordism theories. The coefficient groups are, taking one case as ... Page ii

**denote**the cohomology theories based on the Spectra ason. U l. Exterior algebra We fix in this section a complex inner product Space W of dimension n, and we also fix a unit vector o- e A“w. If n = 4k + 2, we make the exterior algebra ... Page vii

If X e A V and Y e AW then A G. Gy) = (-1)" /,000 A, x) where A , A-1, /* 2

If X e A V and Y e AW then A G. Gy) = (-1)" /,000 A, x) where A , A-1, /* 2

**denote**the maps of section l for A (W 4 W) = AV (9/\W, AV, AW respectively. Proof. Fix an orthonormal basis el, ''', e. for W and °mol” ''' ''mon for W such ...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