INTRODUCTION TO TOPOLOGY AND GEOMETRY PURE AND APPLIED MATHEMATICS A Wiley Series of Texts, Monographs, and Tracts Founded by RICHARD COURANT Editors Emeriti: MYRON B. ALLEN III, DAVID A. COX, PETER HILTON, HARRY HOCHSTADT, PETER LAX, JOHN TOLAND A complete list of the titles in this series appears at the end of this volume. INTRODUCTION TO TOPOLOGY AND GEOMETRY Second Edition Saul Stahl Department of Mathematics The University of Kansas Lawrence, KS Catherine Stenson Department of Mathematics Juniata College Huntington, PA WILEY A JOHN WILEY & SONS, INC., PUBLICATION Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved. Published simultaneously in Canada. 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Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Stahl, Saul. Introduction to topology and geometry. — 2nd edition / Saul Stahl, University of Kansas, Catherine Stenson, Juniata College. pages cm. — (Pure and applied mathematics) Includes bibliographical references and index. ISBN 978-1-118-10810-9 (hardback) 1. Topology 2. Geometry. I. Stenson, Catherine, 1972- II. Title. QA611.S814 2013 514—dc23 2012040259 Printed in the United States of America. 10 9 8 7 6 5 4 32 To Denise, with love from Saul To my family, with love from Cathy. CONTENTS Preface xi Acknowledgments xv 1 Informal Topology 1 Graphs 13 2.1 Nodes and Arcs 13 2.2 Traversability 16 2.3 Colorings 21 2.4 Planarity 25 2.5 Graph Homeomorphisms 31 Surfaces 41 3.1 Polygonal Presentations 42 3.2 Closed Surfaces 50 3.3 Operations on Surfaces 71 3.4 Bordered Surfaces 79 3.5 Riemann Surfaces 94 vii CONTENTS Graphs and Surfaces 103 4.1 Embeddings and Their Regions 103 4.2 Polygonal Embeddings 113 4.3 Embedding a Fixed Graph 118 4.4 Voltage Graphs and Their Coverings 128 Appendix 141 Knots and Links 143 5.1 Preliminaries 144 5.2 Labelings 147 5.3 From Graphs to Links and on to Surfaces 158 5.4 The Jones Polynomial 169 5.5 The Jones Polynomial and Alternating Diagrams 187 5.6 Knots and Surfaces 194 The Differential Geometry of Surfaces 205 6.1 Surfaces, Normals, and Tangent Planes 205 6.2 The Gaussian Curvature 212 6.3 The First Fundamental Form 219 6.4 Normal Curvatures 229 6.5 The Geodesic Polar Parametrization 236 6.6 Polyhedral Surfaces I 242 6.7 Gauss's Total Curvature Theorem 247 6.8 Polyhedral Surfaces II 252 Riemann Geometries 259 Hyperbolic Geometry 275 8.1 Neutral Geometry 275 8.2 The Upper Half-plane 287 8.3 The Half-Plane Theorem of Pythagoras 295 8.4 Half-Plane Isometries 305 The Fundamental Group 317 9.1 Definitions and the Punctured Plane 317 9.2 Surfaces 325 9.3 3-Manifolds 332 9.4 The Poincaré Conjecture 357 CONTENTS IX 10 General Topology 361 10.1 Metric and Topological Spaces 361 10.2 Continuity and Homeomorphisms 367 10.3 Connectedness 377 10.4 Compactness 379 11 Polytopes 387 11.1 Introduction to Polytopes 387 11.2 Graphs of Polytopes 400 11.3 Regular Polytopes 404 11.4 Enumerating Faces 415 Appendix A Curves 429 A. 1 Parametrization of Curves and Arclength 429 Appendix B A Brief Survey of Groups 441 B. 1 The General Background 441 B.2 Abelian Groups 446 B.3 Group Presentations 447 Appendix C Permutations 457 Appendix D Modular Arithmetic 461 Appendix E Solutions and Hints to Selected Exercises 465 References and Resources 497 Index 505 PREFACE This book is intended to serve as a text for a two-semester undergraduate course in topology and modern geometry. It is devoted almost entirely to the geometry of the last two centuries. In fact, some of the subject matter was discovered only within the last two decades. Much of the material presented here has traditionally been part of the realm of graduate mathematics, and its presentation in undergraduate courses necessitates the adoption of certain informalities that would be unacceptable at the more advanced levels. Still, all of these informalities either were used by the mathematicians who created these disciplines or else would have been accepted by them without any qualms. The first four chapters aim to serve as an introduction to topology. Chapter 1 provides an informal explanation of the notion of homeomorphism. This naive intro- duction is in fact sufficient for all the subsequent chapters. However, the instructor who prefers a,more rigorous treatment of basic topological concepts such as homeo- morphisms, topologies, and metric spaces will find it in Chapter 10. The second chapter emphasizes the topological aspects of graph theory, but is not limited to them. This material was selected for inclusion because the accessible na- ture of some of its results makes it the pedagogically perfect vehicle for the transition from the metric Euclidean geometry the students encountered in high school to the combinatorial thinking that underlies the topological results of the subsequent chap- ters. The focal issue here is planarity: Euler's Theorem, coloring theorems, and the Kuratowski Theorem. xi XÜ PREFACE Chapter 3 presents the standard classifications of surfaces of both the closed and bordered varieties. The Euler-Poincaré equation is also proved. Chapter 4 is concerned with the interplay between graphs and surfaces—in other words, graph embeddings. In particular, a procedure is given for settling the question of whether a given graph can be embedded on a given surface. Polygonal (2-cell) embeddings and their rotation systems are discussed. The notion of covering surfaces is introduced via the construction of voltage graphs. The theory of knots and links has recently received tremendous boosts from the work of John Con way, Vaughan Jones, and others. Much of this work is easily accessible, and some has been included in Chapter 5: the Con way-Gordon-Sachs Theorem regarding the intrinsic linkedness of the graph Κ$ in R3 and the invariance of the Jones polynomial. While this discipline is not, properly speaking, topological, connections to the topology of surfaces are not lacking. Knot theory is used to prove the nonembeddability of nonorientable surfaces in R3, and surface theory is used to prove the nondecomposability of trivial knots. The more traditional topic of labelings is also presented. The next three chapters deal with various aspects of differential geometry. The exposition is as elementary as the author could make it and still meet his goals: explanations of Gauss's Total Curvature Theorem and hyperbolic geometry. The ge- ometry of surfaces in R3 is presented in Chapter 6. The development follows that of Gauss's General Investigations of Curved Surfaces. The subtopics include Gaus- sian curvature, geodesies, sectional curvatures, the first fundamental form, intrinsic geometry, and the Total Curvature Theorem, which is Gauss's version of the famed Gauss-Bonnet formula. Some of the technical lemmas are not proved but are instead supported by informal arguments that come from Gauss's monograph. A consider- able amount of attention is given to polyhedral surfaces for the pedagogical purpose of motivating the key theorems of differential geometry. The elements of Riemannian geometry are presented in Chapter 7: Riemann met- rics, geodesies, isometries, and curvature. The numerous examples are also meant to serve as a lead-in to the next chapter. The eighth chapter deals with hyperbolic geometry. Neutral geometry is defined in terms of Euclid's axiomatization of geometry and is described in terms of Euclid's first 28 propositions. Various equivalent forms of the parallel postulates are proven, as well as the standard results regarding the sum of the angles of a neutral triangle. Hyperbolic geometry is also defined axiomatically. Poincaré's half-plane geometry is developed in some detail as an instance of the Riemann geometries of the previous chapter and is demonstrated to be hyperbolic. The isometries of the half-plane are described both algebraically and geometrically. The ninth chapter is meant to serve as an introduction to algebraic topology. The requisite group theory is summarized in Appendix B. The focus is on the derivation of fundamental groups, and the development is based on Poincaré's own exposition and makes use of several of his examples. The reader is taught to derive presentations for the fundamental groups of the punctured plane, closed surfaces, 3-manifolds, and knot complements. The chapter concludes with a discussion of the Poincaré Conjecture.