Table of Contents

Preface

1 Smooth manifolds and smooth maps 1

Tangent spaces and derivatives 2

Regular values 7

The fundamental theorem of algebra 8

2 The theorem of Sard and Brown 10

Manifolds with boundary 12

The Brouwer fixed point theorem 13

3 Proof of Sard's theorem 16

4 The degree modulo 2 of a mapping 20

Smooth homotopy and smooth isotopy 20

5 Oriented manifolds 26

The Brouwer degree 27

6 Vector fields and the Euler number 32

7 Framed cobordism; the Pontryagin construction 42

The Hopf theorem 50

8 Exercises 52

App Classifying 1-manifolds 55

Bibliography 59

Index 63