Introduction to Topology and Geometry
ISBN: 9781118108109 | 1118108108
Edition: 2ndFormat: Hardcover
Publisher: Wiley
Pub. Date: 3/11/2013
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Presenting upper graduate level material in an accessible way for undergraduates, this Second Edition strikes a welcome balance between academic rigor and accessibility while covering an unparalleled range of topics, including the elements of projective geometry, conics and the applications and properties of conic selections, cross ratio points of infinity and fundamental transformations of projective geometry, the points of a homography/involution, and more. In addition, this comprehensive book includes numerous exercises and historical notes.... MORE
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
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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
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.
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