## Characteristic ClassesThe theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds. In this volume, the authors provide a thorough introduction to characteristic classes, with detailed studies of Stiefel-Whitney classes, Chern classes, Pontrjagin classes, and the Euler class. Three appendices cover the basics of cohomology theory and the differential forms approach to characteristic classes, and provide an account of Bernoulli numbers. Based on lecture notes of John Milnor, which first appeared at Princeton University in 1957 and have been widely studied by graduate students of topology ever since, this published version has been completely revised and corrected. |

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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 |