Differential Topology 0821851934, 9780821851937

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DIFFERENTIAL TOPOLOGY VICTOR GUILLEMIN ALAN POLLACK

AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island

DIFFERENTIAL TOPOLOGY

http://dx.doi.org/10.1090/chel/370

DIFFERENTIAL TOPOLOGY VICTOR GUILLEMIN ALAN POLLACK

AMS CHELSEA PUBLISHING American Mathematical Society # Providence, Rhode Island

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2000 Mathematics Subject Classification. Primary 53Cxx, 57Rxx, 58Axx.

For additional information and updates on this book, visit www.ams.org/bookpages/chel-370

Library of Congress Cataloging-in-Publication Data Guillemin, Victor, 1937– Differential topology / Victor Guillemin, Alan Pollack. p. cm. Originally published: Englewood Cliffs, N.J. : Prentice-Hall, 1974. Includes bibliographical references and index. ISBN 978-0-8218-5193-7 (alk. paper) 1. Differential topology. I. Pollack, Alan, 1946– II. Title. QA613.6.G84 2010 514.72—dc22 2010011053

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 1974 held by the American Mathematical Society. All rights reserved.  Reprinted by the American Mathematical Society, 2010, 2014. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2

19 18 17 16 15 14

Prefaces

, xi xv xvii

vi

CONTENTS

CHAPTER

2 Transversality and Intersection

11 11

MANIFOLDS WITH BOUNDARY TRANSVERSALITY

16

THE BORSUK-ULAM THEOREM

2

77

WINDING NUMBERS AND THE JORDAN-BROUWER SEPARATION

85 91

3 Oriented Intersection Theory

11 11 13 14

64

67

INTERSECTION THEORY MOD

THEOREM

CHAPTER

57

ONE-MANIFOLDS AND SOME CONSEQUENCES

13 14 15

57

MOTIVATION ORIENTATION

94

94 95

ORIENTED INTERSECTION NUMBER LEFSCHETZ FIXED-POINT THEORY

107 119

15 16

THE HOPF DEGREE THEOREM

17

THE EULER CHARACTERISTIC AND TRIANGULATIONS

CHAPTER

VECTOR FIELDS AND THE POINCARE-HoPF THEOREM

4 Integration on Manifolds

11

INTRODUCTION

§l

EXTERIOR ALGEBRA

13 14

DIFFERENTIAL FORMS

151 153 162

INTEGRATION ON MANIFOLDS

15 16

EXTERIOR DERIVATIVE

17

STOKES THEOREM

18 19

THE GAuss-BONNET THEOREM

165

174

COHOMOLOGY WITH FORMS

178

182

INTEGRATION AND MAPPINGS

132

141

188 194

151

148

vii

Contents

APPENDIX

1

Measure Zero and Sard's Theorem APPENDIX

202

2

Classification of Compact One-Manifolds

Bibliography Index

217

212

208

Preface to the AMS Chelsea Edition We are deeply grateful to the AMS for reissuing Differential Topology as part of its AMS Chelsea Book Series. Our elementary introduction to topology via transversality techniques has managed to stay in print for most of the thirtysix years since its original appearance, and we would like to thank Edward Dunne and his colleagues in Providence for ensuring its continuing availability (knock on wood) for the next thirty-six years. The techniques it highlights have, in some sense, a very 1970’s flavor. The quixotic hopes of that decade, that singularity theory and catastrophe theory (of whose catastrophic demise the less said the better) would have a revolutionary impact on physics, chemistry, biology, economics, game theory, and investment strategies in the stock market, have proved largely unfounded. However, we have been pleased to find that our students today are, just as were the students of three decades ago, happy with the visceral, down-to-earth approach to topology espoused by books like ours and Milnor’s wonderful Topology from a Differential Viewpoint. We hope (again knock on wood) that whatever the fashions in mathematics of the next thirty-six years, this will continue to be the case. Victor Guillemin Alan Pollack

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http://dx.doi.org/10.1090/chel/370/01

CHAPTER

1 Manifolds and Smooth Maps

§1

Definitions

Beautiful, deep insights into the structure and properties of many geometric spaces can be developed intuitively with the aid of a few tools from elementary calculus. Because calculus is built on the local geometry of Euclidean space, it most naturally adapts itself to spaces that locally "look the same" as some Euclidean space. We call such objects manifolds, spaces in which the environment of each point is "just like" a small piece of Euclidean space. The most familiar examples of manifolds are smooth surfaces like the sphere or torus (the crust of a doughnut), where each point lies in a little curved disk that may be gently flattened into a disk in the plane (Figure I-I). An old friend that does not qualify as a manifold is the cone. Every point but one has a nice Euclidean environment, but no neighborhood of the vertex point looks like a simple piece of the plane. (See Figure 1-2.) To translate our idea into a mathematical definition, we need to make precise the criterion of "sameness." We do so in terms of mappings. A mapping f of an open set U c: Rn into RIn is called smooth if it has continuous partial derivatives of all orders. However, when the domain off is not open, one usually cannot speak of partial derivatives. (Why 1) So we adapt the open situation to more general spaces. A map f: X -- RIn defined on an 1

§I

3

Definitions

where U is an open set ofRft. So smoothness is a local property;!: X - 0 Rm is smooth if it is smooth in a neighborhood of each point of X. (Contrasting with "local" is the term global, which refers to the whole space X as a unified object.) A smooth map! : X - 0 Y of subsets of two Euclidean spaces is a diffeomorphism ifit is one to one and onto, and if the inverse map/-I: Y - 0 Xis also smooth. X and Yare diffeomorphic if such a map exists. In our philosophy, two diffeomorphic sets are intrinsically equivalent. They may be considered as two copies of a single abstract space, which may happen to be differently situated in their surrounding Euclidean spaces. Soon you will develop sufficient intuition to recognize easiJy many spaces as diffeomorphic. Perhaps you can begin by thinking about a few pictorial examples (Figure 1-4).

•

• Closed interval

Diffeomorphic

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L8 Not diffeomorphic

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