Introduction to Topological Manifolds
This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and relate
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John M. Lee
Introduction to Topological Manifolds Second Edition
John M. Lee Department of Mathematics University of Washington Seattle, Washington 98195-4350 USA [email protected]
Editorial Board: S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
K. A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720 USA [email protected]
ISSN 0072-5285 ISBN 978-1-4419-7939-1 e-ISBN 978-1-4419-7940-7 DOI 10.1007/978-1-4419-7940-7 Springer New York Dordrecht Heidelberg London Mathematics Subject Classification (2010): 54-01, 55-01, 57-01 © 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, U.S.A.), 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)
Preface
Manifolds are the mathematical generalizations of curves and surfaces to arbitrary numbers of dimensions. This book is an introduction to the topological properties of manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. Here at the University of Washington, for example, this text is used for the first third of a year-long course on the geometry and topology of manifolds; the remaining two-thirds of the course focuses on smooth manifolds using the tools of differential geometry. There are many superb texts on general and algebraic topology available. Why add another one to the catalog? The answer lies in my particular vision of graduate education: it is my (admittedly biased) belief that every serious student of mathematics needs to be intimately familiar with the basics of manifold theory, in the same way that most students come to know the integers, the real numbers, vector spaces, functions of one real or complex variable, groups, rings, and fields. Manifolds play a role in nearly every major branch of mathematics (as I illustrate in Chapter 1), and specialists in many fields find themselves using concepts and terminology f
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