Characteristic Classes. (AM-76)

Description

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 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