## Characteristic ClassesThe theory of characteristic classes began in the year 1935 with almost simultaneous work by Hassler Whitney in the United States and Eduard Stiefel in Switzerland. Stiefel's thesis, written under the direction of Heinz Hopf, introduced and studied certain 'characteristic' homology classes determined by the tangent bundle of a smooth manifold. |

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#### Review: Characteristic Classes. (Am-76)

User Review - Jamie - GoodreadsI just didn't get it. Maybe I didn't invest myself into it enough. Davis and Kirk say that every mathematician should read this book. Whatever (ie I disagree). Read full review

#### Review: Characteristic Classes. (Am-76)

User Review - Dustin Tran - GoodreadsSpring 2012: Math 215B - Homotopy Theory & Characteristic Classes; Denis Auroux. Ch. 1-14, along with Hatcher's AT. Very well laid-out and easy to understand. The notation is a bit old and the book ... Read full review

### Contents

Preface 1 Smooth Manifolds | 3 |

2 Vector Bundles | 13 |

3 Constructing New Vector Bundles Out of Old | 25 |

4 StiefelWhitney Classes | 37 |

5 Grassmann Manifolds and Universal Bundles | 55 |

6 A Cell Structure for Grassmann Manifolds | 73 |

7 The Cohomology Ring HGn Z2 | 83 |

8 Existence of StiefelWhitney Classes | 89 |

15 Pontrjagin Classes | 173 |

16 Chern Numbers and Pontrjagin Numbers | 183 |

17 The Oriented Cobordism Ring fl | 199 |

18 Thom Spaces and Transversality | 205 |

19 Multiplicative Sequences and the Signature Theorem | 219 |

20 Combinatorial Pontrjagin Classes | 231 |

Epilogue | 249 |

Singular Homology and Cohomology | 257 |

9 Oriented Bundles and the Euler Class | 95 |

10 The Thom Isomorphism Theorem | 105 |

11 Computations in a Smooth Manifold | 115 |

12 Obstructions | 139 |

13 Complex Vector Bundles and Complex Manifolds | 149 |

14 Chern Classes | 155 |

### Common terms and phrases

algebra base space basis bundle f bundle map canonically isomorphic characteristic classes Chern class cobordism cohomology class cohomology group Compare completes the proof complex manifold complex vector bundle compute construct COROLLARY corresponding cross-section CW-complex defined DEFINITION denote diffeomorphism dimension direct limit equal Euclidean metric Euler class exact sequence example fiber F finite follows easily formula given Gn(Rn+k Grassmann manifold Hence Hn(E Hn(M homology homomorphism identity induced LEMMA Let f line bundle linearly map f matrix modulo multiplicative sequence neighborhood non-zero normal bundle open set open subset orthogonal paracompact partition piecewise linear Pontrjagin classes Pontrjagin numbers power series Problem projection map projective space prove real vector bundle regular value Riemannian satisfies smooth manifold smooth map Spanier Steenrod Stiefel Stiefel-Whitney classes Stiefel-Whitney numbers structure symmetric tangent bundle theorem Thom topological total space trivial unique vector space Whitney sum zero