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Mostly Surfaces Richard Evan Schwartz Mostly Surfaces Richard Evan Schwartz Department of Mathematics, Brown University Current address: 151 Thayer St. Providence, RI 02912 E-mail address: [email protected] 2010 Mathematics Subject Classification. Primary Key words and phrases. surfaces, geometry, topology, complex analysis supported by N.S.F. grant DMS-0604426. Contents Preface xiii Chapter 1. Book Overview 1 1.1. Behold, the Torus! 1 § 1.2. Gluing Polygons 3 § 1.3.DrawingonaSurface 5 § 1.4. Covering Spaces 8 § 1.5. HyperbolicGeometryandtheOctagon 9 § 1.6. Complex Analysis and Riemann Surfaces 11 § 1.7. ConeSurfacesandTranslationSurfaces 13 § 1.8. TheModularGroupandtheVeechGroup 14 § 1.9. Moduli Space 16 § 1.10. Dessert 17 § Part 1. Surfaces and Topology Chapter2. DefinitionofaSurface 21 2.1. A Word about Sets 21 § 2.2. Metric Spaces 22 § 2.3. OpenandClosedSets 23 § 2.4. Continuous Maps 24 § v vi Contents 2.5. Homeomorphisms 25 § 2.6. Compactness 26 § 2.7. Surfaces 26 § 2.8. Manifolds 27 § Chapter3. TheGluingConstruction 31 3.1. GluingSpacesTogether 31 § 3.2. TheGluingConstructioninAction 34 § 3.3. The Classification of Surfaces 36 § 3.4. TheEulerCharacteristic 38 § Chapter 4. The Fundamental Group 43 4.1. A Primer on Groups 43 § 4.2. Homotopy Equivalence 45 § 4.3. The Fundamental Group 46 § 4.4. Changing the Basepoint 48 § 4.5. Functoriality 49 § 4.6. Some First Steps 51 § Chapter 5. Examples of Fundamental Groups 53 5.1.TheWindingNumber 53 § 5.2. The Circle 56 § 5.3. The Fundamental Theorem of Algebra 57 § 5.4. The Torus 58 § 5.5. The 2-Sphere 58 § 5.6. TheProjectivePlane 59 § 5.7. A Lens Space 59 § 5.8. The Poincar´eHomology Sphere 62 § Chapter6. CoveringSpacesandtheDeckGroup 65 6.1. Covering Spaces 65 § 6.2. The Deck Group 66 § 6.3. A Flat Torus 67 § Contents vii 6.4. More Examples 69 § 6.5. SimplyConnectedSpaces 70 § 6.6. The Isomorphism Theorem 71 § 6.7. The Bolzano–Weierstrass Theorem 72 § 6.8. TheLiftingProperty 73 § 6.9. ProofoftheIsomorphismTheorem 74 § Chapter7. ExistenceofUniversalCovers 79 7.1. The Main Result 80 § 7.2. The Covering Property 82 § 7.3. SimpleConnectedness 84 § Part 2. Surfaces and Geometry Chapter8. EuclideanGeometry 89 8.1. Euclidean Space 89 § 8.2. The Pythagorean Theorem 92 § 8.3. The X Theorem 93 § 8.4. Pick’s Theorem 94 § 8.5. The Polygon Dissection Theorem 98 § 8.6. Line Integrals 100 § 8.7. Green’s Theorem for Polygons 102 § Chapter9. SphericalGeometry 105 9.1. Metrics,TangentPlanes,andIsometries 105 § 9.2. Geodesics 107 § 9.3. Geodesic Triangles 109 § 9.4. Convexity 111 § 9.5. StereographicProjection 113 § 9.6. The Hairy Ball Theorem 114 § Chapter 10. Hyperbolic Geometry 117 10.1. Linear Fractional Transformations 117 § 10.2. CirclePreservingProperty 118 § viii Contents 10.3. The Upper Half-Plane Model 120 § 10.4. Another Point of View 123 § 10.5. Symmetries 124 § 10.6. Geodesics 126 § 10.7. The Disk Model 127 § 10.8. Geodesic Polygons 129 § 10.9. Classification of Isometries 133 § Chapter11. RiemannianMetricsonSurfaces 135 11.1. Curves in the Plane 135 § 11.2. RiemannianMetricsonthePlane 136 § 11.3. Diffeomorphisms and Isometries 137 § 11.4. Atlases and Smooth Surfaces 138 § 11.5. Smooth Curves and the Tangent Plane 139 § 11.6. Riemannian Surfaces 141 § 11.7. Riemannian Covers 143 § Chapter12. HyperbolicSurfaces 147 12.1. Definition 147 § 12.2. Gluing Recipes 150 § 12.3. GluingRecipesLeadtoSurfaces 151 § 12.4. Some Examples 154 § 12.5. Geodesic Triangulations 154 § 12.6. Hadamard’s Theorem 157 § 12.7. The Hyperbolic Cover 159 § 12.8. Orientations 162 § Part 3. Surfaces and Complex Analysis Chapter 13. A Primer on Complex Analysis 167 13.1. Basic Definitions 167 § 13.2. Cauchy’s Theorem 170 § 13.3. The Cauchy Integral Formula 171 § Contents ix 13.4. Differentiability 172 § 13.5. The Maximum Principle 174 § 13.6. Removable Singularities 175 § 13.7. Power Series 176 § 13.8. Taylor Series 178 § Chapter 14. Disk and Plane Rigidity 181 14.1. Disk Rigidity 181 § 14.2. Liouville’s Theorem 184 § 14.3. StereographicProjectionRevisited 185 § Chapter 15. The Schwarz–Christoffel Transformation 189 15.1. The Basic Construction 190 § 15.2. The Inverse Function Theorem 191 § 15.3. Proof of Theorem 15.1 192 § 15.4. The Range of Possibilities 194 § 15.5. Invariance of Domain 196 § 15.6. TheExistenceProof 197 § Chapter 16. Riemann Surfaces and Uniformization 203 16.1. Riemann Surfaces 203 § 16.2. Maps Between Riemann Surfaces 205 § 16.3. The Riemann Mapping Theorem 207 § 16.4. The Uniformization Theorem 210 § 16.5. The Small Picard Theorem 211 § 16.6. Implications for Compact Surfaces 212 § Part 4. Flat Cone Surfaces Chapter 17. Flat Cone Surfaces 217 17.1. Sectors and Euclidean Cones 217 § 17.2. Euclidean Cone Surfaces 218 § 17.3. The Gauss–Bonnet Theorem 219 § 17.4. Translation Surfaces 221 § x Contents 17.5. Billiards and Translation Surfaces 222 § 17.6. Special Maps on a Translation Surface 226 § 17.7. ExistenceofPeriodicBilliardPaths 228 § Chapter 18. Translation Surfaces and the Veech Group 231 18.1. Affine Automorphisms 231 § 18.2. The Differential Representation 233 § 18.3. Hyperbolic Group Actions 234 § 18.4. Proof of Theorem 18.1 236 § 18.5. Triangle Groups 238 § 18.6. LinearandHyperbolicReflections 239 § 18.7. Cylinders and Dehn Twists 242 § 18.8. Behold, The Double Octagon! 244 § Part 5. The Totality of Surfaces Chapter 19. Continued Fractions 251 19.1. The Gauss Map 251 § 19.2. Continued Fractions 253 § 19.3. The Farey Graph 254 § 19.4. StructureoftheModularGroup 256 § 19.5. Continued Fractions and the Farey Graph 257 § 19.6. The Irrational Case 259 § Chapter20. Teichm¨ullerSpaceandModuliSpace 263 20.1. Parallelograms 263 § 20.2. Flat Tori 264 § 20.3. The Modular Group Again 266 § 20.4. Moduli Space 268 § 20.5. Teichm¨uller Space 270 § 20.6. The Mapping Class Group 272 § Chapter 21. Topology of Teichm¨uller Space 275 21.1. Pairs of Pants 275 § Contents xi 21.2. Pants Decompositions 277 § 21.3. Special Maps and Triples 279 § 21.4. The End of the Proof 281 § Part 6. Dessert Chapter 22. The Banach–Tarski Theorem 287 22.1. The Result 287 § 22.2. TheSchroeder–BernsteinTheorem 288 § 22.3. The Doubling Theorem 290 § 22.4. Depleted Balls 291 § 22.5. The Depleted Ball Theorem 291 § 22.6. A Result about Polynomials 294 § 22.7. The Injective Homomorphism 295 § Chapter23. Dehn’sDissectionTheorem 297 23.1. The Result 297 § 23.2. Dihedral Angles 298 § 23.3. Irrationality Proof 299 § 23.4. Rational Vector Spaces 300 § 23.5. Dehn’s Invariant 301 § 23.6. Clean Dissections 302 § 23.7. The Proof 304 § Chapter 24. The Cauchy Rigidity Theorem 305 24.1. The Main Result 305 § 24.2. The Dual Graph 306 § 24.3. OutlineoftheProof 307 § 24.4. Proof of Lemma 24.3 308 § 24.5. The Cauchy Arm Lemma 311 § 24.6. Proof of Lemma 24.2 312 § 24.7. The Loss of Strict Convexity 314 § 24.8. Proof of the Cauchy Arm Lemma 316 § xii Contents Bibliography 319 Index 321 Preface This book is based on notes I wrote when teaching an undergraduate seminar on surfaces at Brown University in 2005. Each week I wrote up notes on a different topic. Basically, I told the students about many of the great things I have learned about surfaces over the years. I tried to do things in as direct a fashion as possible, favoring concrete results over a buildup of theory. Originally, I had written 14 chapters, but later I added 9 more chapters so as to make a more substantial book. Each chapter has its own set of exercises. The exercises are em- bedded within the text. Most of the exercises are fairly routine, and advance the arguments being developed, but I tried to put a few challenging problems in each batch. If you are willing to accept some results on faith, it should be possible for you to understand the mate- rial without working the exercises. However, you will get much more out of the book if you do the exercises. The central object in the book is a surface. I discuss surfaces from many points of view: as metric spaces, triangulated surfaces, hyperbolic surfaces, and so on. The book has many classical results about surfaces, both geometric and topological, and it also has some extraneous stuff that I included because I like it. For instance, the book contains proofs of the Pythagorean Theorem, Pick’s Theorem, xiii xiv Preface Green’s Theorem, Dehn’s Dissection Theorem, the Cauchy Rigidity Theorem, and the Fundamental Theorem of Algebra. All the material in the book can be found in various textbooks, though there probably isn’t one textbook that has it all. Whenever possible, I will point out textbooks or other sources where you can read more about what I am talking about. The various fields of math surrounding the concept of a surface—geometry, topology, complex analysis, combinatorics—are deeply intertwined and often related in surprising ways. I hope to present this tapestry of ideas in a clear and rigorous yet informal way. My general view of mathematics is that most of the complicated things we learn have their origins in very simple examples and phe- nomena. A good way to master a body of mathematics is to first understand all the sources that lead to it. In this book, the square torus is one of the key simple examples. A great deal of the the- ory of surfaces is a kind of elaboration of phenomena one encounters when studying the square torus. In the first chapter of the book, I will introduce the square torus and describe the various ways that its structure can be modified and generalized. I hope that this first chapter serves as a good guide to the rest of the book.
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