INTRODUCTION TO TOPOLOGY AND GEOMETRY

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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

W I L E Y A JOHN WILEY & SONS, INC., PUBLICATION

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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 3 2

To Denise, with love from Saul

To my family, with love from Cathy.

CONTENTS

Preface xi

Acknowledgments xv

1 Informal Topology 1

Graphs 2.1 Nodes and Arcs 2.2 Traversability 2.3 Colorings 2.4 Planarity 2.5 Graph Homeomorphisms

Surfaces

3.1 Polygonal Presentations 3.2 Closed Surfaces 3.3 Operations on Surfaces 3.4 Bordered Surfaces 3.5 Riemann Surfaces

13

13 16 21 25 31

41

42 50 71 79 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.

PREFACE XÜi

The tenth chapter serves a dual purpose. On the one hand it aims to acquaint the reader with the elegant topic of general topology and the joys of sequence chasing. On the other hand, it contains the rigorous definitions of a variety of fundamental concepts that were only informally defined in the previous chapters. In terms of mathematical maturity, this is probably the most demanding part of the book.

The last chapter is devoted to the study of polytopes. Following an introduction, attention is given to the graphs of polytopes, regular polytopes, and the enumeration of faces of polytopes.

Wherever appropriate, historical notes have been interspersed with the exposi-tion. Care was taken to supply many exercises that range from the routine to the challenging. Middle-level exercises were hard to come by, and the author welcomes all suggestions.

An Instructor's solution manual is available upon request from Wiley.

S A U L S T A H L

Lawrence, Kansas

Stahlex@ku.edu

ACKNOWLEDGMENTS

The first author is deeply indebted to Mark Hunacek for reading the manuscript and suggesting many improvements. Jack Porter helped by making his Graduate Topol-ogy Notes and other materials available to me. He also corrected some errors in the early version of Chapter 10. Encouraging kind words and valuable criticisms were provided by the reviewers Michael J. Kallaher, David Royster, Dan Gottlieb, and David W. Henderson, as well as several others who chose to remain anonymous. Stephen Quigley and Susanne Steitz-Filler at John Wiley and Sons supervised the conversion of the notes into a book. The manuscript was expertly typeset by Larisa Martin and Sandra Reed. Prentice Hall gave me its gracious permission to reprint portions of Chapters 9-12 of my book Geometry from Euclid to Knots, which it published in 2003.

S.S.

xv

xvi

The second author is grateful to Saul Stahl for the invitation to join this project and for his help throughout. Thanks to Marge Bayer for recommending me and for her comments on a draft, and to Lou Billera for his comments and for his guidance over the years. Thanks also to the Mathematics Institute of Leiden University for their hospitality in the spring of 2012.

C. S.

CHAPTER 1

INFORMAL TOPOLOGY

In this chapter the notion of a topological space is introduced, and informal ad hoc methods for identifying equivalent topological spaces and distinguishing between nonequivalent ones are provided.

The last book of Euclid's opus Elements is devoted to the construction of the five Platonic solids pictured in Figure 1.1. A fact that Euclid did not mention is that the counts of the vertices, edges, and faces of these solids satisfy a simple and elegant relation. If these counts are denote by v, e, and / , respectively, then

v - e + / = 2. (1)

Specifically, for these solids we have:

Cube: Octahedron: Tetrahedron:

Dodecahedron: Icosahedron:

8 - 1 2 + 6 = 2. 6 - 1 2 + 8 = 2. 4 - 6 + 4 = 2. 2 0 - 3 0 + 1 2 = 1 2 - 3 0 + 20 =

Introduction to Topology and Geometry, Second Edition. 1 By Saul Stahl and Catherine Stenson Copyright © 2013 John Wiley & Sons, Inc.

2 INFORMAL TOPOLOGY

% 4 Cube Tetrahedron

Octahedron

Dodecahedron Icosahedron

Figure 1.1 The Platonic solids.

A Platonic solid is defined by the specifications that each of its faces is the same regular polygon and that the same number of faces meet at each vertex. An inter-esting feature of Equation (1) is that while the Platonic solids depend on the notions of length and straightness for their definition, these two aspects are absent from the equation itself. For example, if each of the edges of the cube is either shrunk or extended by some factor, whose value may vary from edge to edge, a lopsided cube is obtained (Fig. 1.2) for which the equation still holds by virtue of the fact that it holds for the (perfect) cube. This is also clearly true for any similar modification of the other four Platonic solids. The fact of the matter is that Equation (1) holds

Figure 1.2 A lopsided cube.

not only for distorted Platonic solids, but for all solids as well, provided these solids are carefully defined. Thus, for the three solids of Figure 1.3 we have respectively 5 - 8 + 5 = 2, 6 - 9 + 5 = 2, and 7 - 12 + 7 = 2. The applicability of Equation (1) to all such solids was first noted by Leonhard Euler (1707-1783) in 1758, although some historians contend that this equation was presaged by certain observations of René Descartes (1596-1650).

Euler's equation remains valid even after the solids are subjected to a wider class of distortions which result in the curving of their edges and faces (see Figure 1.4). One need simply relax the definition of edges and faces so as to allow for any non-self intersecting curves and surfaces. Soccer balls and Volleyballs, together with the patterns formed by their seams, are examples of such curved solids to which Euler's equation applies. Moreover, it is clear that the equation still holds after the balls are deflated.

Topology is the study of those properties of geometrical figures that remain valid even after the figures are subjected to distortions. This is commonly expressed by saying that topology is rubber-sheet geometry. Accordingly, our necessarily infor-mal definition of a topological space identifies it as any subset of space from which the notions of straightness and length have been abstracted; only the aspect of conti-guity remains. Points, arcs, loops, triangles, solids (both straight and curved), and the surfaces of the latter are all examples of topological spaces. They are, of course, also geometrical objects, but topology is only concerned with those aspects of their ge-

Figure 1.3 Three solids.

4 INFORMAL TOPOLOGY

Figure 1.4 A curved cube.

ometry that remain valid despite any translations, elongations, inflations, distortions, or twists.

Another topological problem investigated by Euler, somewhat earlier, in 1736, is known as the bridges ofKoenigsberg. At that time this Prussian city straddled the two banks of a river and also included two islands, all of which were connected by seven bridges in the pattern indicated in Figure 1.5. On Sunday afternoons the citizens of Koenigsberg entertained themselves by strolling around all of the city's parts, and eventually the question arose as to whether an excursion could be planned which would cross each of the seven bridges exactly once. This is clearly a geometrical problem in that its terms are defined visually, and yet the exact distances traversed in such excursions are immaterial (so long as they are not excessive, of course). Nor are the precise contours of the banks and the islands of any consequence. Hence, this is a topological problem. Theorem 2.2.2 will provide us with a tool for easily resolving this and similar questions.

The notorious Four-Color Problem, which asks whether it is possible to color the countries of every geographical map with four colors so that adjacent countries sharing a border of nonzero length receive distinct colors, is also of a topological nature. Maps are clearly visual objects, and yet the specific shapes and sizes of the countries in such a map are completely irrelevant. Only the adjacency patterns matter.

Every mathematical discipline deals with objects or structures, and most will pro-vide a criterion for determining when two of these are identical, or equivalent. The equality of real numbers can be recognized from their decimal expansions, and two vectors are equal when they have the same direction and magnitude. Topological equivalence is called homeomorphism. The surface of a sphere is homeomorphic to

Figure 1.5 The city of Koenigsberg.

Figure 1.6 Homeomorphic open arcs.

those of a cube, a hockey puck, a plate, a bowl, and a drinking glass. The reason for this is that each of these objects can be deformed into any of the others. Sim-ilarly, the surface of a doughnut is homeomorphic to those of an inner tube, a tire, and a coffee mug. On the other hand, the surfaces of the sphere and the doughnut are not homeomorphic. Our intuition rejects the possibility of deforming the sphere into a doughnut shape without either tearing a hole in it or else stretching it out and juxtaposing and pasting its two ends together. Tearing, however, destroys some con-tiguities, whereas juxtaposition introduces new contiguities where there were none before, and so neither of these transformations is topologically admissible. This in-tuition of the topological difference between the sphere and the doughnut will be confrmed by a more formal argument in Chapter 3.

The easiest way to establish the homeomorphism of two spaces is to describe a deformation of one onto the other that involves no tearing or juxtapositions. Such a deformation is called an isotopy. Whenever isotopies are used in the sequel, their existence will be clear and will require no formal justification. Such is the case, for instance, for the isotopies that establish the homeomorphisms of all the open arcs in Figure 1.6, all the loops in Figure 1.7, and all the ankh-like configurations of Figure 1.8. Note that whereas the page on which all these curves are drawn is two-dimensional, the context is definitely three-dimensional. In other words, all our curves (and surfaces) reside in Euclidean 3-space M3, and the isotopies may make use of all three dimensions.

The concept of isotopy is insufficient to describe all homeomorphisms. There are spaces which are homeomorphic but not isotopic. Such is the case for the two loops in Figure 1.9. It is clear that loop b is isotopic to all the loops of Figure 1.7 above, and it is plausible that loop a is not, a claim that will be justified in Chapter 5. Hence,

Figure 1.7 Homeomorphic loops.

6 INFORMAL TOPOLOGY

Figure 1.8 Homeomorphic ankhs.

Figure 1.9 Two spaces that are homeomorphic but not isotopic.

the two loops are not isotopic to each other. Nevertheless, they are homeomorphic in the sense that ants crawling along these loops would experience them in identical manners. To express this homeomorphism somewhat more formally it is necessary to resort to the language of functions. First, however, it should be pointed out that the word function is used here in the sense of an association, or an assignment, rather than the end result of an algebraic calculation. In other words, a function / : S -^ T is simply a rule that associates to every point of S a point of T. In this text most of the functions will be described visually rather than algebraically.

Given two topological spaces S and Γ, a homeomorphism is a function / : S -^ T such that

1. / matches all the points of S to all the points of T (distinct points of S are matched with distinct points of T and vice versa);

2. / preserves contiguity.

It is the vagueness of the notion of contiguity that prevents this from being a formal definition. Since any two points on a line are separated by an infinitude of other points, this concept is not well defined. The homeomorphism of S and T is denoted by S « T. The homeomorphism of the loops of Figure 1.9 can now be established by orienting them, labeling their lowest points A and B, and matching points that are at equal distances from A and B, where the distance is measured along the oriented loop (Fig. 1.10). Of course, the positions of A and B can be varied without affecting the existence of the homeomorphism.

A similar function can be defined so as to establish the homeomorphism of any two loops as long as both are devoid of self-intersections. Suppose two such loops c and uf, of lengths y and δ respectively, are given (Fig. 1.11). Again begin by specifying orientations and initial points C and D on the two loops. Then, for every

7

real number 0 < r < 1, match the point at distance ry from C along c with the point at distance rS from D along d.

Figure 1.12 contains another instructive example. Each of its three topological spaces consists of a band of, say, width 1 and length 20. They differ in that e is untwisted, / has one twist, and g has two twists. Band / differs from the other two in that its border is in fact one single loop whereas bands e and g have two distinct borders each. It therefore comes as no surprise that band / is not homeomorphic to either e or g. These last two, however, are homeomorphic to each other. To describe this homeomorphism a coordinate system is established on each of the bands as follows. For each number 0 < r < 1 let Ler and Lgr denote the oriented loops of length 20 that run along the band at a constant distance r from the bottom borders of e and g respectively. Choosing start lines as described in Figure 1.13, the coordinate pair (r,s), 0 < r < 1, 0 < s < 20, describes those points on the loops Ler and L^r at a distance s from the respective starting line. The required homeomorphism simply matches up points of e and g that have the same coordinate pairs. The reason this wouldn't work for band / is that for this band the coordinatization process fails (see Fig. 1.14).

As mentioned above, / is not homeomorphic to e and g, because it has a different number of borders. In general, borders and other extremities are a good place to look for differences between topological spaces. For example, every two of the spaces in Figure 1.15 are nonhomeomorphic because they each have a different number of extremities. The number of components of a space can also serve as a tool for distinguishing between homeomorphism types. All the spaces in Figure 1.16 have the same number of extremities, but they are nevertheless nonhomeomorphic because each has a different number of components: 1, 2, 3, and 4, respectively.

Another method for distinguishing between spaces is to examine what remains when an equal number of properly selected points are deleted from each. For in-stance, both spaces of Figure 1.17 have one component, and neither has extremities. Nevertheless, they are not homeomorphic, because the removal of the two endpoints of the diameter of the θ-like space results in a space with three components, whereas the removal of any two points of the circle leaves only two components. In general a topological property of a space is a property that is shared by all the spaces that are homeomorphic to it. The number of endpoints and the number of components are

Q fiQ)

A B

Figure 1.10 A homeomorphism of two loops.

8 INFORMAL TOPOLOGY

.750

C D

Figure 1.11 A homeomorphism of two loops.

Figure 1.12 Three bands.

Start here.

Start here.

Figure 1.13 The homeomorphism of two bands.

Figure 1.14 A failed homeomorphism.

Figure 1.15 Four nonhomeomorphic spaces.

both such topological properties. On the other hand, neither the length of an interval nor the area of a region is a topological property.

10 INFORMAL TOPOLOGY

7K N/

Figure 1.16 Four nonhomeomorphic spaces.

Figure 1.17 Two nonhomeomorphic spaces.

The foregoing discussion of topological spaces, homeomorphisms, and isotopies is an informal working introduction that will serve for the purposes of this text. Ex-perience indicates that this lack of precision will not hamper the comprehension of the subsequent material. Rigorous definitions are provided in Chapter 10, which can be read out of sequence.

In working out the exercises, the readers may find it useful to note that both home-omorphism and isotopy are equivalence relations in the sense that they satisfy the following three conditions:

Reflexivity: Every topological space is homeomorphic (isotopic) to

itself.

Symmetry: If 5 is homeomorphic (isotopic) to Γ, then T is homeo-

morphic (isotopic) to S.

Transitivity: If R is homeomorphic (isotopic) to S and S is homeo-

morphic (isotopic) to Γ, then R is homeomorphic

(isotopic) to T.

Exercises 1.1

1. Which of the letters in Figure 1.18 are homeomorphic? 2. Which of the topological spaces in Figure 1.19 are homeomorphic? 3. Which of the topological spaces in Figure 1.20 are homeomorphic? 4. Which of the topological spaces of Figure 1.21 are isotopic? 5. Are the following statements true or false? Justify your answers.

(a) If two topological spaces are homeomorphic, then they are also isotopic.

11

(b) If two topological spaces are isotopic, then they are also homeomorphic.

(c) Topological equivalence is synonymous with homeomorphism.

(d) Topological equivalence is synonymous with isotopy.

(e) Every two loops are isotopic.

(f) Every two loops are homeomorphic.

A B C D E F G - H I J K L M N O P Q R S T U V W X Y Z

Figure 1.18 Twenty-six topological spaces.

(H) © (KD (HD

(3D Φ'Φ OD 0^0

Figure 1.19 Some one-dimensional topological spaces.

AX ΔΑ a b e d e

Figure 1.20 Some one-dimensional topological spaces.

ΌΟ

12 INFORMAL TOPOLOGY

0-0 ® GO § 3 ξ Figure 1.21 Some one-dimensional topological spaces.