Characteristic Classes

Front Cover
Princeton University Press, 1974 - Mathematics - 330 pages

The 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.

 

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 HG₁ Z2
83
8 Existence of StiefelWhitney Classes
89
15 Pontrjagin Classes
173
16 Chern Numbers and Pontrjagin Numbers
183
17 The Oriented Cobordism Ring
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
Bernoulli Numbers
281
Connections Curvature and Characteristic Classes
289
Bibliography
315
Index
325
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