Universitext Editorial Board (North America). S.Axler K.A.Ribet
Universitext Editorial Board (North America): S. Axler K.A. Ribet
Loring W.Tu An Introduction to Manifolds Springer
Loring W. Tu An Introduction to Manifolds
Loring W.Tu Medford,MA 02155 loring.tu@tufts.edu N) K.A.Ribet Praisc C Berkeley.CA 94720-3840 at Berkeley USA USA axler@sfsu.cdu ribet@math.berkeley.edu ISBN-13:978-0-387-48098-5 e-SBN-13:978-0-387-48101-2 Mathematics Classification Code(2000):58-01.58Axx.58A05.58A10,58A12 Library of Congress Control Number:2007932203 008 Springer Science+Business Media.LLC. NY 10013.US subjectoproprietary rights Printed onacid-free paper. 987654321 www.springer.com (JLS/MP)
Loring W. Tu Department of Mathematics Tufts University Medford, MA 02155 loring.tu@tufts.edu Editorial Board (North America): S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA axler@sfsu.edu K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA ribet@math.berkeley.edu ISBN-13: 978-0-387-48098-5 e-ISBN-13: 978-0-387-48101-2 Mathematics Classification Code (2000): 58-01, 58Axx, 58A05, 58A10, 58A12 Library of Congress Control Number: 2007932203 © 2008 Springer Science + Business Media, LLC. Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 www.springer.com (JLS/MP) 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 identifi ed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, All rights reserved. This work may not be translated or copied in whole or in part without the written
Dedicated to the memory of Raoul Bott
Dedicated to the memory of Raoul Bott
Preface It has been more than two decades since Raoul Bott and Ipublished Differential Forms in Algebraic Topology.While this book has enjoyed a certain success,it does assume some familiarity with manifolds and so is not so readily accessible to the average first-year graduate student in mathematics.It has been my goal for quite some time to bridge this gap by writing an elementary introduction to manifolds assuming only one semester of abstract algebra and a ve ear of real analysis.Moreover,given the emen hetw een t ot only bu mathematicians and advanced undergraduates,but also physicists who want a solid foundation in geometry and topology. With so many excellent books on manifolds on the market,any author who un- dertakes to write another owes to the public,if not to himself,a good rationale.First and foremost is my desire to write a readable but rigorous introduction that gets the reader quickly up to speed,to the point where for example he or she can compute de rham cohomolooy of simple s A secondc nside in the prerequ Most books labo a as a su a Euclidean space.This has the disadvantage of making quotie manifolds,of which a projective space is a prime example,difficult to understand. My solution is to make the first four chapters of the book independent of point-set topology and to place the necessary point-set topology in an appendix.While reading the first four chapters,the student should at the same time study Appendix A to acquire the point-set topology that will be assumed starting in Chapter 5. The book is meant to be read and studied by a novice.It is not meant to be encvclopedic.Therefore.I discuss only the irreducible minimum of manifold theor which I think eve ery math allows the centra deas to emerge more clearly.In several years of teaching.I hav generally been able to cover the entire book in one semester. In order not to interrupt the flow of the exposition,certain proofs of a more routine or computational nature are left as exercises.Other exercises are scattered throughout the exposition,in their natural context.In addition to the exercises embedded in the
Preface It has been more than two decades since Raoul Bott and I published Differential Forms in Algebraic Topology. While this book has enjoyed a certain success, it does assume some familiarity with manifolds and so is not so readily accessible to the average first-year graduate student in mathematics. It has been my goal for quite some time to bridge this gap by writing an elementary introduction to manifolds assuming only one semester of abstract algebra and a year of real analysis. Moreover, given the tremendous interaction in the last twenty years between geometry and topology on the one hand and physics on the other, my intended audience includes not only budding mathematicians and advanced undergraduates, but also physicists who want a solid foundation in geometry and topology. With so many excellent books on manifolds on the market, any author who undertakes to write another owes to the public, if not to himself, a good rationale. First and foremost is my desire to write a readable but rigorous introduction that gets the reader quickly up to speed, to the point where for example he or she can compute de Rham cohomology of simple spaces. A second consideration stems from the self-imposed absence of point-set topology in the prerequisites. Most books laboring under the same constraint define a manifold as a subset of a Euclidean space. This has the disadvantage of making quotient manifolds, of which a projective space is a prime example, difficult to understand. My solution is to make the first four chapters of the book independent of point-set topology and to place the necessary point-set topology in an appendix. While reading the first four chapters, the student should at the same time studyAppendixAto acquire the point-set topology that will be assumed starting in Chapter 5. The book is meant to be read and studied by a novice. It is not meant to be encyclopedic. Therefore, I discuss only the irreducible minimum of manifold theory which I think every mathematician should know. I hope that the modesty of the scope allows the central ideas to emerge more clearly. In several years of teaching, I have generally been able to cover the entire book in one semester. In order not to interrupt the flow of the exposition, certain proofs of a more routine or computational nature are left as exercises. Other exercises are scattered throughout the exposition, in their natural context. In addition to the exercises embedded in the