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 StiefelWhitney 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. (Am76)
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. (Am76)
User Review  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
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 