Lectures on Morse Homology Kluwer Texts in the Mathematical Sciences

VOLUME 29

A Graduate-Level Book Series

The titles published in this series are listed at the end of this volume. SPRINGER SCIENCE+BUSINESS MEDIA, B.V. Lectures on Morse Homology

by Augustin Banyaga The Pennsylvania State University, University Park, PA, U.S.A. and David Hurtubise The Pennsylvania State University, Altoona, P.A., US.A.

SPRINGER SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-6705-0 ISBN 978-1-4020-2696-6 (eBook) DOI 10.1007/978-1-4020-2696-6

Printed on acid-free paper

All Rights Reserved © 2004 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004 Softcover reprint of the hardcover 1st edition 2004 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Contents

Preface lX 1. Introduction 1 1.1 Overview 1 1.2 Algebraic topology 3 1.3 Basic Morse theory 4 1.4 Stable and unstable manifolds 5 1.5 Basic differential topology 6 1.6 Morse-Smale functions 7 1.7 The Morse Homology Theorem 9 1.8 Morse theory on Grassmann manifolds 10 1.9 Floer homology theories 11 1.10 Guide to the book 11 2. The CW-Homology Theorem 15 2.1 Singular homology 15 2.2 Singular cohomology 20 2.3 CW-complexes 21 2.4 CW-homology 23 2.5 Some homotopy theory 31 3. Basic Morse Theory 45 3.1 Morse functions 45 3.2 The gradient flow of a Morse function 58 3.3 The CW-complex associated to a Morse function 63 3.4 The Morse Inequalities 73 VI Lectures on Morse Homology

3.5 Morse-Bott functions 80 4. The Stable/Unstable Manifold Theorem 93 4.1 The Stable/Unstable Manifold Theorem for a Morse function 93 4.2 The Local Stable Manifold Theorem 98 4.3 The Global Stable/Unstable Manifold Theorem 111 4.4 Examples of stable/unstable manifolds 116 5. Basic Differential Topology 127 5.1 Immersions and submersions 127 5.2 Transversality 131 5.3 Stability 132 5.4 General position 134 5.5 Stability and density for Morse functions 137 5.6 Orientations and intersection numbers 143 5.7 The Lefschetz Fixed Point Theorem 148 6. Morse-Smale Functions 157 6.1 The Morse-Smale transversality condition 157 6.2 The ..\-Lemma 165 6.3 Consequences ofthe ..\-Lemma 171 6.4 The CW-complex associated to a Morse-Smale function 175 7. The Morse Homology Theorem 195 7.1 The Morse-Smale-Witten boundary operator 196 7.2 Examples using the Morse Homology Theorem 201 7.3 The Conley index 207 7.4 Proof of the Morse Homology Theorem 211 7.5 Independence of the choice ofthe index pairs 219 8. Morse Theory On Grassmann Manifolds 227 8.1 Morse theory on the adjoint orbit of a Lie group 228 8.2 A Morse function on an adjoint orbit of the unitary group 235 8.3 An almost complex structure on the adjoint orbit 243 8.4 The critical points and indices offA : U (n + k) · xo ----+ IR: 246 8.5 A Morse function on the complex Grassmann manifold 249 8.6 The gradient flow lines offA : Gn,n+k (

8.8 Further generalizations and applications 260 9. An Overview of Floer Homology Theories 269 9.1 Introduction to Floer homology theories 269 9.2 Symplectic Floer homology 272 9.3 Floer homology for Lagrangian intersections 280 9.4 Instanton Floer homology 281 9.5 A symplectic flavor of the instanton homology 284 Hints and References for Selected Problems 287 Bibliography 309 Symbol Index 317 Index 321 Preface

This book is based on the lecture notes from a course we taught at Penn State University during the fall of 2002. The main goal of the course was to give a complete and detailed proof of the Morse Homology Theorem (Theo• rem 7.4) at a level appropriate for second year graduate students. The course was designed for students who had a basic understanding of singular homol• ogy, CW-complexes, applications of the existence and uniqueness theorem for O.D.E.s to vector fields on smooth Riemannian manifolds, and Sard's Theo• rem.

We would like to thank the following students for their participation in the course and their help proofreading early versions of this manuscript: James Barton, Shantanu Dave, Svetlana Krat, Viet-Trung Luu, and Chris Saunders. We would especially like to thank Chris Saunders for his dedication and en• thusiasm concerning this project and the many helpful suggestions he made throughout the development of this text.

We would also like to thank Bob Wells for sharing with us his extensive knowledge of CW-complexes, Morse theory, and singular homology. Chapters 3 and 6, in particular, benefited significantly from the many insightful conver• sations we had with Bob Wells concerning a Morse function and its associated CW-complex.

Augustin Banyaga and David Hurtubise The Pennsylvania State University, 2004