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 ixAcknowledgments xiii1 Informal Topology 12 Graphs 132.1 Nodes and Arcs 132.2 Traversability 162.3 Colorings 212.4 Planarity 252.5 Graph Homeomorphisms 313 Surfaces 413.1 Polygonal Presentations 423.2 Closed Surfaces 503.3 Operations on Surfaces 713.4 Bordered Surfaces 793.5 Riemann Surfaces 944 Graphs and Surfaces 1034.1 Embeddings and Their Regions 1034.2 Polygonal Embeddings 1134.3 Embedding a Fixed Graph 1184.4 Voltage Graphs and Their Coverings 128Appendix: 1415 Knots and Links 1435.1 Preliminaries 1445.2 Labelings 1475.3 From Graphs to Links and on to Surfaces 1585.4 The Jones Polynomial 1695.5 The Jones Polynomial and Alternating Diagrams 1875.6 Knots and surfaces 1946 The Differential Geometry of Surfaces 2056.1 Surfaces, Normals, and Tangent Planes 2056.2 The Gaussian Curvature 2126.3 The First Fundamental Form 2196.4 Normal Curvatures 2296.5 The Geodesic Polar Parametrization 2366.6 Polyhedral Surfaces I 2426.7 Gauss’s Total Curvature Theorem 2476.8 Polyhedral Surfaces II 2527 Riemann Geometries 2598 Hyperbolic Geometry 2758.1 Neutral Geometry 2758.2 The Upper Half Plane 2878.3 The HalfPlane Theorem of Pythagoras 2958.4 HalfPlane Isometries 3059 The Fundamental Group 3179.1 Definitions and the Punctured Plane 3179.2 Surfaces 3259.3 3Manifolds 3329.4 The Poincar´e Conjecture 35710 General Topology 36110.1 Metric and Topological Spaces 36110.2 Continuity and Homeomorphisms 36710.3 Connectedness 37710.4 Compactness 37911 Polytopes 38711.1 Introduction to Polytopes 38711.2 Graphs of Polytopes 40111.3 Regular Polytopes 40511.4 Enumerating Faces 415Appendix A Curves 429A.1 Parametrization of Curves and Arclength 429Appendix B A Brief Survey of Groups 441B.1 The General Background 441B.2 Abelian Groups 446B.3 Group Presentations 447Appendix C Permutations 457Appendix D Modular Arithmetic 461Appendix E Solutions and Hints to Selected Exercises 465References and Resources 497