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Differential Manifolds This Is Volume 138 in PURE and APPLIED MATHEMATICS Differential Manifolds This is Volume 138 in PURE AND APPLIED MATHEMATICS H. Bass, A. Borel, J. Moser, and S.-T. Yau, editors Paul A. Smith and Samuel Eilenberg, founding editors Differential Manifolds Antoni A. Kosinski Department of Mathematics Rutgers University New Brunswick, New Jersey ACADEMIC PRESS, INC. Harcourt Brace Jovanovich, Publishers Boston San Diego New York London Sydney Tokyo Toronto This book is printed on acid-free paper. @ Copyright 0 1993 by Academic Press, Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. ACADEMIC PRESS, INC. 1250 Sixth Avenue, San Diego, CA 92101 United Kingdom edition published by ACADEMIC PRESS LIMITED 24-28 Oval Road, London NW1 7DX Library of Congress Cataloging-in-Publication Data Kosinski, Antoni A.. date. Differential manifolds / Antoni A. Kosinski. p. cm. Includes bibliographical references (p. ) and index. ISBN 0-12-421850-4 (acid-free paper) 1. Differentiable manifolds. I. Title. QA614.3.K668 1992 5 14.36~20 91-34981 CIP Printed in the United States of America 92939495 BB 9 8 7 6 5 4 3 2 1 This book is dedicated to the memory of my father, the founder and publisher of "Mathesis," the first scientific publishing house in Poland. April 1963 Symposium in honor of Marston Morse, Institute for Advanced Study, Princeton, New Jersey. First row, seated,j?om left: S. S. Chem, R. J. Pohrer, A. Selberg, M. Morse,W. Leighton, M. Hirsch, S. S. Caims, H. Whitney. Second row, standing, from left: R. Bott, B. Mazur, G. A. Hedlund, T. Frankel, S. Smale, N. Kuiper, J. F. Adams, W. Browder, J. W. Milnor, M. Kervaire. Contents Introduction xi Notational Conventions xv I. Differentiable Structures 1 1. Smooth Manifolds and Maps 1 2. Partitions of Unity 6 3. Smooth Vector Bundles 8 4. Tangent Space 12 5. Vector Fields 16 6. Differential Equations on a Smooth Manifold 18 7. Collars 21 11. Immersions, Imbeddings, Submanifolds 25 1. Local Equivalence of Maps 25 2. Submanifolds 26 3. Imbeddings in R" 32 4. Isotopies 33 5. Ambient Isotopies 36 6. Historical Remarks 38 111. Normal Bundle, Tubular Neighborhoods 41 1. Exponential Map 41 2. Normal Bundle and Tubular Neighborhoods 44 vii viii CONTENTS 3. Uniqueness of Tubular Neighborhoods 49 4. Submanifolds of the Boundary 52 5. Inverse Image of a Regular Value 55 6. The group r" 56 7. Remarks 57 IV. Transversality 59 1. Transversal Maps and Manifolds 59 2. Transversality Theorem 63 3. Morse Functions 66 4. Neighborhood of a Critical Point 68 5. Intersection Numbers 70 6. Historical Remarks 73 V. Foliations 75 1. &Fields 76 2. Foliations 78 3. Frobenius Theorem 80 4. Leaves of a Foliation 82 5. Examples 84 VI. Operations on Manifolds 89 1. Connected Sum 90 2. # and Homotopy Spheres 94 3. Boundary Connected Sum 97 4. Joining Manifolds along Submanifolds 99 5. Joining Manifolds along Submanifolds of the Boundary 100 6. Attaching Handles 103 7. Cancellation Lemma 106 8. Combinatorial Attachment 110 9. Surgery 112 10. Homology and Intersections in a Handle 113 11. (m,k)-Handlebodies, m > 2k 115 12. (2k, k)-Handlebodies; Plumbing 118 VII. Handle Presentation Theorem 125 1. Elementary Cobordisms 125 2. Handle Presentation Theorem 127 3. Homology Data of a Cobordism 13 1 4. Morse Inequalities 135 5. Poincark Duality 136 6. 0-Dimensional Handles 137 7. Heegaard Diagrams 138 8. Historical Remarks 141 CONTENTS ix VIII. The h-Cobordism Theorem 143 1. Elementary Row Operations 144 2. Cancellation of Handles 148 3. 1-Handles 151 4. Minimal Presentation; Main Theorems 152 5. h-cobordism; The Group 8" 156 6. Highly Connected Manifolds 159 7. Remarks 161 IX. Framed Manifolds 167 1. Framings 168 2. Framed Submanifolds 171 3. Ok(M") 174 4. OO(M") 177 5. The Pontriagin Construction 179 6. Operations on Framed Submanifolds and Homotopy Theory 183 7. v-Manifolds 186 8. Almost Parallelizable Manifolds 189 9. Historical Remarks 192 X. Surgery 195 1. Effect of Surgery on Homology 197 2. Framing a Surgery; Surgery below Middle Dimension 200 3. Surgery on 4n-Dimensional Manifolds 202 4. Surgery on (4n + 2)-Dimensional Manifolds 206 5. Surgery on Odd-Dimensional Manifolds 210 6. Computation of 8" 215 7. Historical Note 219 Appendix 223 1. Implicit Function Theorem 223 2. A Lemma of M. Morse 226 3. Brown-Sard Theorem 226 4. Orthonormalization 227 5. Homotopy Groups of SO(k) 230 Bibliography 233 Index 24 1 This Page Intentionally Left Blank Introduction Three decades ago differential topology went through a period of extremely rapid growth. Six years after Milnor’s discovery of non-equivalent smooth structures on a topological sphere such structures were already classified through the newly invented method of surgery. In the same period Smale showed that every manifold can be constructed by successive attachment of handles and provided a method to obtain the most economical descrip- tion. This enabled him to prove the PoincarC conjecture in higher dimensions, thereby demonstrating the strength of his methods. Also in the same period, Smale, Haefliger, and Hirsch developed the theory of imbed- dings and immersions, vastly extending the foundational results of Whitney. The methods invented in these early years subsequently gave rise to a large amount of research. Thus the handle constructions of Smale were extended by Kirby and Siebenmann to topological manifolds, and the method of surgery was expanded and applied successfully to a large variety of problems by Browder, Novikov, and Wall. Looking back at these developments it appears possible at present to single out certain main ideas and results and present them in a systematic and consolidated way. This is what I have attempted to do here. In a very broad way, the content of this book can be described as the study of the topological structure of smooth manifolds. As I intended this book to be xi xii INTRODUCTION accessible at the beginning of graduate studies, this is preceded by a presentation of basic concepts and tools of differential topology. An overview of the material follows. More precise information is in the introductions to the individual chapters. The first two chapters introduce in moderate detail the notions of smooth manifold, submanifold, and tangent space. Imbeddings are discussed briefly and isotopies at length. The presentation is complete, but it is assumed, implicitly, that the subject is not totally unfamiliar to the reader. A more leisurely treatment of the analytical topics of these two chapters can be found in [Bo]. In Chapter 111 it is shown that a neighborhood of a submanifold of a smooth manifold can be fibered by planes, that is, it is a vector bundle, and that this bundle structure is unique. This is a fundamental result and the basis for all that follows. The concept of transversality due to Thom is introduced in Chapter IV. This is the smooth counterpart of the notion of general position and is used similarly to extract from messy entanglements their essential geometric content. In this chapter it is applied to prove that every function can be approximated by one with a very regular behavior at singularities, a Morse function. It is also used to define intersection numbers. This geometric concept will in later chapters supplant the less intuitive cup product. The results of Chapter V are not utilized elsewhere in this book. It provides an introduction to the beautiful and difficult theory of foliations. These first four, or five, chapters constitute a general background not only for differential topology but also for the study of Lie groups and Riemannian manifolds. The analytical means employed here have their roots in the implicit function theorem, the theory of ordinary differential equations, and the Brown-Sard Theorem. Some algebraic results in the form adapted for the purpose and collected in the appendix are used as well. Very little algebraic topology enters the picture at this stage. Chapter VI is devoted to a description of various ways of gluing manifolds together: connected sum, connected sum along the boundary, attachment of handles, etc. The presentation avoids the usual smoothing of comers. There is a brief discussion of the effect of these operations on homology; it prepares the ground for the more precise results of the following chapters. The last two sections describe a way to build some highly connected manifolds; it is shown in Chapter VIII that all highly connected manifolds can be constructed in this way. INTRODUCTION xiii An important result of Chapter VI describes the situation when two successive attachments of handles produce no change: The second handle destroys the first. This is Smale’s Cancellation Lemma. Chapter VII begins with the proof that every manifold can be built by a successive attachment of handles of increasing dimension. To such a struc- ture there is associated a chain complex yielding the homology of the manifold. The chains are linear combinations of handles and the boundary operator is given by a matrix of intersection numbers. Of course, the same is true for a triangulation or a cellular decomposition, but the relation between the handle presentation and the homology structure of the manifold is very transparent geometrically. This fact is exploited here to obtain a simple proof of the PoincarC duality theorem and of the Morse inequalities providing the lower limit for the number of handles necessary to construct a manifold with given homology. The 3-dimensional case, at the end of the chapter, provides a nice illustration of basic ideas. Chapter VIII contains the proof of the existence of a handle presentation with the minimal number of handles determined by its homology groups.
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