Characteristic Classes_John W.Milnor, James D.Stasheff.pdf

Characteristic Classes

John W.Milnor

James D.Stasheff

2023, paralogism@mail.ustc.edu.cn

Contents

Contents 2

Preface 1

1 Smooth Manifolds 2

2 Vector Bundles 8

n

8 Existence of Stiefel-Whitney Classes 55

9 Oriented Bundles and the Euler Class 59

10 The Thom Isomorphism Theorem 64

11 Computations in a Smooth Manifold 70

12 Obstructions 83

13 Complex Vector Bundles and Complex Manifolds 89

14 Chern Classes 92

15 Pontrjagin Classes 103

16 Chern Numbers and Pontrjagin Numbers 109

Epilogue 147

Preface

The text which follows is based mostly on lectures at Princeton University in 1957. The senior author

wihses to apologize for the delay in publication.

The theory of characteristic classes began in the year 1935 with almost simultaneous work by Hassler

Whitney in the United States and Eduard Stiefel in Switzeland. Stiefel’s thesis, written under the direction

of Heinz Hopf, introduced and studied certain “characteristic” homology classes determined by the

tangent bundle of a smooth manifold. Whitney, then at Harvard University, treated the case of an arbitrary

sphere bundle. Somewhat later he invented the language of cohomology theory, hence the concept of a

characteristic cohomology class, and proved the basic product theorem.

complex Grassmann manifolds have a cohomology structure which is much easier to understand than

that of the real Grassmann manifolds. This has led to a great clarification of the theory of real characteristic

classes.

matically active: Whitney at the Institute for Advanced Study in Princeton, Stiefel as director of the Institute

for Applied Mathematics of the Federal Institute of Technology in Zürich, Pontrjagin as director of the

Steklov Institute in Moscow, and Chern at the University of California in Berkeley. This book is delicated to

them.

be denoted by R.

The word “smooth” will be used as a synonym for “differentialbel of class

C

∞

”. Thus a function defined

on an open set

n

with values in

R

k

is smooth if its partial derivatives of all orders exist and are

continuous.

R

:

A

We topologize this space

R

A

as a Cartesian product of copies of

R

. For any subset

M ⊂R

A

, we give

functions f

Y →R is continuous. Here Y can be arbitrary topological space.

U →R

A

for i =1, . . . , n.

smooth function

h

:

U →R

A

defined on an open set U ⊂R

n

such that

Of course our previous coordinate space

R

can be obtained as a special case of this more general concept, simply by taking

A

to

be the set of integers between 1 and n.