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Winding Around the Winding Number in Topology, Geometry, and Analysis STUDENT MATHEMATICAL LIBRARY Volume 76 Winding Around The Winding Number in Topology, Geometry, and Analysis John Roe HEMAT MAT ICS STUDY A DVANCED S EMESTERS American Mathematical Society Mathematics Advanced Study Semesters Winding Around The Winding Number in Topology, Geometry, and Analysis http://dx.doi.org/10.1090/stml/076 STUDENT MATHEMATICAL LIBRARY Volume 76 Winding Around The Winding Number in Topology, Geometry, and Analysis John Roe HEMAT MAT ICS STUDY A DVANCED S EMESTERS American Mathematical Society Mathematics Advanced Study Semesters Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 55M25; Secondary 55M05, 47A53, 58A10, 55N15. For additional information and updates on this book, visit www.ams.org/bookpages/stml-76 Library of Congress Cataloging-in-Publication Data Roe, John, 1959– Winding around : the winding number in topology, geometry, and analysis / John Roe. pages cm. — (Student mathematical library ; volume 76) Includes bibliographical references and index. ISBN 978-1-4704-2198-4 (alk. paper) 1. Mathematical analysis—Foundations. 2. Associative law (Mathematics) 3. Symmetric functions. 4. Commutative law (Mathematics) I. Title. QA299.8.R64 2015 515—dc23 2015019246 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http:// www.ams.org/rightslink. Send requests for translation rights and licensed reprints to reprint-permission @ams.org. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2015 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 201918171615 Contents Foreword: MASS and REU at Penn State University ix Preface xi Chapter 1. Prelude: Love, Hate, and Exponentials 1 §1.1. Two sets of travelers 1 §1.2. Winding around 5 §1.3. The most important function in mathematics 6 §1.4. Exercises 12 Chapter 2. Paths and Homotopies 15 §2.1. Path connectedness 15 §2.2. Homotopy 18 §2.3. Homotopies and simple-connectivity 22 §2.4. Exercises 25 Chapter 3. The Winding Number 27 §3.1. Maps to the punctured plane 27 §3.2. The winding number 29 §3.3. Computing winding numbers 33 §3.4. Smooth paths and loops 38 v vi Contents §3.5. Counting roots via winding numbers 42 §3.6. Exercises 46 Chapter 4. Topology of the Plane 49 §4.1. Some classic theorems 49 §4.2. The Jordan curve theorem I 54 §4.3. The Jordan curve theorem II 59 §4.4. Inside the Jordan curve 63 §4.5. Exercises 67 Chapter 5. Integrals and the Winding Number 73 §5.1. Differential forms and integration 73 §5.2. Closed and exact forms 79 §5.3. The winding number via integration 84 §5.4. Homology 87 §5.5. Cauchy’s theorem 94 §5.6. A glimpse at higher dimensions 95 §5.7. Exercises 97 Chapter 6. Vector Fields and the Rotation Number 101 §6.1. The rotation number 101 §6.2. Curvature and the rotation number 105 §6.3. Vector fields and singularities 107 §6.4. Vector fields and surfaces 113 §6.5. Exercises 117 Chapter 7. The Winding Number in Functional Analysis 121 §7.1. The Fredholm index 121 §7.2. Atkinson’s theorem 125 §7.3. Toeplitz operators 129 §7.4. The Toeplitz index theorem 133 §7.5. Exercises 136 Contents vii Chapter 8. Coverings and the Fundamental Group 139 §8.1. The fundamental group 139 §8.2. Covering and lifting 143 §8.3. Group actions 150 §8.4. Examples 153 §8.5. The Nielsen-Schreier theorem 157 §8.6. An application to nonassociative algebra 161 §8.7. Exercises 165 Chapter 9. Coda: The Bott Periodicity Theorem 169 §9.1. Homotopy groups 169 §9.2. The topology of the general linear group 174 Appendix A. Linear Algebra 181 §A.1. Vector spaces 181 §A.2. Basis and dimension 184 §A.3. Linear transformations 188 §A.4. Duality 192 §A.5. Norms and inner products 194 §A.6. Matrices and determinants 197 Appendix B. Metric Spaces 203 §B.1. Metric spaces 203 §B.2. Continuous functions 206 §B.3. Compact spaces 208 §B.4. Function spaces 213 Appendix C. Extension and Approximation Theorems 217 §C.1. The Stone-Weierstrass theorem 217 §C.2. The Tietze extension theorem 220 Appendix D. Measure Zero 223 §D.1. Measure zero subsets of R and of S1 223 viii Contents Appendix E. Calculus on Normed Spaces 229 §E.1. Normed vector spaces 229 §E.2. The derivative 231 §E.3. Properties of the derivative 233 §E.4. The inverse function theorem 237 Appendix F. Hilbert Space 239 §F.1. Definition and examples 239 §F.2. Orthogonality 243 §F.3. Operators 246 Appendix G. Groups and Graphs 249 §G.1. Equivalence relations 250 §G.2. Groups 251 §G.3. Homomorphisms 254 §G.4. Graphs 257 Bibliography 261 Index 265 Foreword: MASS and REU at Penn State University This book is part of a collection published jointly by the Amer- ican Mathematical Society and the MASS (Mathematics Advanced Study Semesters) program as a part of the Student Mathematical Library series. The books in the collection are based on lecture notes for advanced undergraduate topics courses taught at the MASS and/or Penn State summer REU (Research Experiences for Under- graduates). Each book presents a self-contained exposition of a non- standard mathematical topic, often related to current research areas, accessible to undergraduate students familiar with an equivalent of two years of standard college mathematics and suitable as a text for an upper division undergraduate course. Started in 1996, MASS is a semester-long program for advanced undergraduate students from across the USA. The program’s curricu- lum amounts to sixteen credit hours. It includes three core courses from the general areas of algebra/number theory, geometry/topology, and analysis/dynamical systems, custom designed every year; an in- terdisciplinary seminar; and a special colloquium. In addition, ev- ery participant completes three research projects, one for each core course. The participants are fully immersed into mathematics, and ix x Foreword: MASS and REU at Penn State University this, as well as intensive interaction among the students, usually leads to a dramatic increase in their mathematical enthusiasm and achieve- ment. The program is unique for its kind in the United States. The summer mathematical REU program is formally indepen- dent of MASS, but there is a significant interaction between the two: about half of the REU participants stay for the MASS semester in the fall. This makes it possible to offer research projects that re- quire more than seven weeks (the length of the REU program) for completion. The summer program includes the MASS Fest, a two to three day conference at the end of the REU at which the partici- pants present their research and that also serves as a MASS alumni reunion. A nonstandard feature of the Penn State REU is that, along with research projects, the participants are taught one or two intense topics courses. Detailed information about the MASS and REU programs at Penn State can be found on the website www.math.psu.edu/mass. Preface Mathematics is an endlessly fruitful subject. One reason is its ability to make lemons into lemonade. In mathematics, the gap between what we’re hoping to prove and what is actually true can itself become something that we can measure, something we can quantify — the basis for a whole new world of mathematical theory. Let me give an example. In Calculus II, you learn that every (rea- sonably smooth) function of one variable is the derivative of another function — the fundamental theorem of calculus says that integration is the reverse of differentiation. In Calculus III, you find out that in higher dimensions there is a necessary condition that must be satis- fied if n given functions are to be the partial derivatives of another function. For instance, in dimension 2, if functions u and v are to be the partial derivatives (with respect to x and y) of a function f,then the integrability condition ∂u ∂v = ∂y ∂x must be satisfied. Is this necessary condition always sufficient? For functions de- fined on a disc, the answer is yes (“every irrotational vector field is the gradient of a potential”). On more general domains, though, the xi xii Preface answer is no, as is shown by the notorious example y −x u = ,v= x2 + y2 x2 + y2 defined on R2 \{(0, 0)}. Most Calculus III courses treat this as a nui- sance, an anomaly. What if we instead treated it as a clue, a signpost, the start of a trail that might lead to a new kind of mathematics? In fact, this trailhead leads us up one of the many routes to the summit of Mount Winding-Number, one of the most beautiful peaks of the Mathematical Range1.
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