## Algebraic and Geometric SurgeryThis book is an introduction to surgery theory: the standard classification method for high-dimensional manifolds. It is aimed at graduate students, who have already had a basic topology course, and would now like to understand the topology of high-dimensional manifolds. This text contains entry-level accounts of the various prerequisites of both algebra and topology, including basic homotopy and homology, Poincare duality, bundles, co-bordism, embeddings, immersions, Whitehead torsion, Poincare complexes, spherical fibrations and quadratic forms and formations. While concentrating on the basic mechanics of surgery, this book includes many worked examples, useful drawings for illustration of the algebra and references for further reading. |

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### Contents

The surgery classification of manifolds | 1 |

Manifolds | 14 |

Homotopy and homology | 29 |

Poincare duality | 48 |

Bundles | 86 |

Cobordism theory | 124 |

Embeddings immersions and singularities | 142 |

Whitehead torsion | 170 |

g Poincare complexes and spherical fibrations | 193 |

Surgery on maps | 218 |

The evendimensional surgery obstruction | 247 |

The odddimensional surgery obstruction | 302 |

The structure set | 340 |

361 | |

367 | |

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### Common terms and phrases

A-module morphism abelian group algebraic surgery BG(k BO(k BO(m bordant BSO(n chain complex chain equivalence chain map Chapter classification cobordism cohomology CW complex Definition degree 1 normal diffeomorphism Dn+1 e-quadratic form element Everg exact sequence Example f.g. free A-modules f.g. projective fc-plane bundle fibration fibre finite CW complex framed n-embedding geometric Poincare complex handle decomposition Hn(M Hn(X homology homotopy class homotopy equivalence homotopy groups Hopf invariant i-cobordism immersion g isomorphism kernel formation killed Kn(M L-groups l)n-quadratic formation lagrangian m-dimensional geometric Poincare m-dimensional manifold map f Milnor morphism n-connected n-connected 2n n-immersion n-surgery nonsingular normal bordism normal map onlg oriented cover Poincare duality Proof Proposition quadratic regular homotopy s-Cobordism Theorem Section self-intersection Sm+k Sn x Dm-n sphere spherical fibrations split e-quadratic stable isomorphism submanifold surgery obstruction surgery theory symmetric form Thom space torsion transverse trivial Umkehr universal cover vector bundle vector space Whitney