An Invitation to Morse Theory
This self-contained treatment of Morse theory focuses on applications and is intended for a graduate course on differential or algebraic topology. The book is divided into three conceptually distinct parts. The first part contains the foundations of Morse
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Universitext Series Editors: Sheldon Axler San Francisco State University Vincenzo Capasso Universit`a degli Studi di Milano Carles Casacuberta Universitat de Barcelona Angus J. MacIntyre Queen Mary, University of London Kenneth Ribet University of California, Berkeley Claude Sabbah ´ CNRS, Ecole Polytechnique Endre S¨uli University of Oxford Wojbor A. Woyczynski Case Western Reserve University
Universitext is a series of textbooks that presents material from a wide variety of mathematical disciplines at master’s level and beyond. The books, often well class-tested by their author, may have an informal, personal even experimental approach to their subject matter. Some of the most successful and established books in the series have evolved through several editions, always following the evolution of teaching curricula, to very polished texts. Thus as research topics trickle down into graduate-level teaching, first textbooks written for new, cutting-edge courses may make their way into Universitext.
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Liviu Nicolaescu
An Invitation to Morse Theory Second Edition
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Liviu Nicolaescu Department of Mathematics University of Notre Dame Notre Dame USA [email protected]
ISSN 0172-5939 e-ISSN 2191-6675 ISBN 978-1-4614-1104-8 e-ISBN 978-1-4614-1105-5 DOI 10.1007/978-1-4614-1105-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011939083 Mathematics Subject Classification (2010): 53A04, 53A05, 53D20, 57R17, 57R58, 57R65, 57R70, 57R91, 58A35, 58K05, 58K10 © Springer Science+Business Media, LLC 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To my mother, with deepest gratitude
Preface
As the title suggests, the goal of this book is to give the reader a taste of the “unreasonable effectiveness” of Morse theory. The main idea behind this technique can be easily visualized. Suppose M is a smooth, compact manifold, which for simplicity we assume is embedded in a Euclidean space E. We would like to understand basic topological invariants of M such as its homology, and we attempt a “slicing” technique. We fix a unit vector u in E and we start slicing M with the family of hyperplanes perpendicular to u. Such a hyperplane will in general intersect M along a subm
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