Characteristic Classes. (AM-76)

✍ Scribed by John Milnor, James D. Stasheff

• Publisher: Princeton University Press
• Year: 1974
• Tongue: English
• Leaves: 333
• Edition: TeXromancers
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Contents
Preface
Preface To The New Typesetting
Smooth Manifolds
Vector Bundles
Euclidean Vector Bundles
Constructing New Vector Bundles Out of Old
Stiefel-Whitney Classes
Consequences of the Four Axioms
Division Algebras
Immersions
Stiefel-Whitney Number
Grassmann Manifold and Universal Bundles
Infinite Grassmann Manifolds
The Universal Bundle gamma n
Paracompact Spaces
Characteristic Classes of Real n-Plane Bundles
A Cell Structure for Grassmann Manifolds
The Cohomology Ring H(Gr n; Z/2
Uniqueness of Stiefel-Whitney Classes
Existence of Stiefel-Whitney Classes
Verification of the Axioms
Oriented Bundles and the Euler Class
The Thom Isomorphism Theorem
Computations in a Smooth Manifold
The Normal Bundle
The Tangent Bundle
The Diagonal Cohomology Class in H n(MxM)
Poincaré Duality and the Diagonal Class
Euler Class and Euler Characteristic
Wu's Formula for Stiefel-Whitney Classes
Obstructions
The Gysin Sequence of a Vector Bundle
The Oriented Universal Bundle
The Euler Class as an Obstruction
Complex Vector Spaces and Complex Manifolds
Chern Classes
Hermitian Metrics
Construction of Chern Classes
Complex Grassmann Manifolds
The Product Theorem for Chern Classes
Dual or Conjugate Bundles
The Tangent Bundle of Complex Projective Space
Pontrjagin Classes
The Cohomology of the Oriented Grassmann Manifold
Chern Numbers and Pontrjagin Numbers
Partitions
Chern Numbers
Pontrjagin Numbers
Symmetric Functions
A Product Formula
Linear Independence of Chern Numbers and of Pontrjagin Numbers
The Oriented Cobordism Ring omega
Smooth Manifolds-with-Boundary
Oriented Cobordism
Thom Spaces and Transitivity
The Thom Space of a Euclidean Vector Bundle
Homotopy Groups Modulo Ab<infinity
Regular Values and Transversality
The Main Theorem
Multiplicative Sequences and the Signature Theorem
Multiplicative Characteristic Classes
Combinatorial Pontrjagin Classes
The Differentiable Case
The Combinatorial Case
Applications
Epilogue
Non-Differentiable Manifolds
Smooth Manifolds with Additional Structure
Generalized Cohomology Theories
Appendices
Singular Homology and Cohomology
Basic Definitions
Editor's notes: Relative (co)homology
The Relationship between Homology and Cohomology
Homology of a CW-Complex
Cup Products
Cohomology of Product Spaces
Homology of Manifolds
The Fundamental Homology Class of a Manifold
Cohomology with Compact Support
The Cap Product Operation
Bernoulli Numbers
Connections, Curvature and Characteristic Classes
Bibliography
Index