Introduction to Topology and Geometry

1.Informal Topology 2.Graphs 2.1.Nodes and Arcs 2.2.Traversability 2.3.Colorings 2.4.Planarity 2.5.Graph Homeomorphisms 3.Surfaces 3.1.Polygonal Presentations 3.2.Closed Surfaces 3.3.Operations on Surfaces 3.4.Bordered Surfaces 3.5.Riemann Surfaces 4.Graphs and Surfaces 4.1.Embeddings and Their Regions 4.2.Polygonal Embeddings 4.3.Embedding a Fixed Graph 4.4.Voltage Graphs and Their Coverings Appendix 5.Knots and Links 5.1.Preliminaries 5.2.Labelings 5.3.From Graphs to Links and on to Surfaces 5.4.The Jones Polynomial 5.5.The Jones Polynomial and Alternating Diagrams 5.6.Knots and Surfaces 6.The Differential Geometry of Surfaces 6.1.Surfaces, Normals, and Tangent Planes 6.2.The Gaussian Curvature 6.3.The First Fundamental Form 6.4.Normal Curvatures 6.5.The Geodesic Polar Parametrization 6.6.Polyhedral Surfaces I 6.7.Gauss's Total Curvature Theorem Contents note continued: 6.8.Polyhedral Surfaces II 7.Riemann Geometries 8.Hyperbolic Geometry 8.1.Neutral Geometry 8.2.The Upper Half-plane 8.3.The Half-Plane Theorem of Pythagoras 8.4.Half-Plane Isometries 9.The Fundamental Group 9.1.Definitions and the Punctured Plane 9.2.Surfaces 9.3.3-Manifolds 9.4.The Poincare Conjecture 10.General Topology 10.1.Metric and Topological Spaces 10.2.Continuity and Homeomorphisms 10.3.Connectedness 10.4.Compactness 11.Polytopes 11.1.Introduction to Polytopes 11.2.Graphs of Polytopes 11.3.Regular Polytopes 11.4.Enumerating Faces Appendix A Curves A.1.Parametrization of Curves and Arclength Appendix B A Brief Survey of Groups B.1.The General Background B.2.Abelian Groups B.3.Group Presentations Appendix C Permutations Appendix D Modular Arithmetic Appendix E Solutions and Hints to Selected Exercises.