Introduction to Topology and Geometry

SAUL STAHL, PhD, is Professor in the Department of Mathematics at the University of Kansas and twice the winner of the Carl B. Allendoerfer Award from the Mathematical Association of America.

CATHERINE STENSON, PhD, is Professor of Mathematics at Juniata College in Huntingdon, Pennsylvania.

Preface ix

Acknowledgments xiii

1 Informal Topology 1

2 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

3 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

4 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

5 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

6 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

7 Riemann Geometries 259

8 Hyperbolic Geometry 275

8.1 Neutral Geometry 275

8.2 The Upper Half Plane 287

8.3 The HalfPlane Theorem of Pythagoras 295

8.4 HalfPlane Isometries 305

9 The Fundamental Group 317

9.1 Definitions and the Punctured Plane 317

9.2 Surfaces 325

9.3 3Manifolds 332

9.4 The Poincar´e Conjecture 357

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 401

11.3 Regular Polytopes 405

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

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