Stahl / Stenson Introduction to Topology and Geometry

2. Auflage 2014
ISBN: 978-1-118-54614-7
Verlag: John Wiley & Sons
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 528 Seiten, E-Book

Reihe: Wiley Series in Pure and Applied Mathematics

An easily accessible introduction to over threecenturies of innovations in geometry
Praise for the First Edition
". . . a welcome alternative to compartmentalizedtreatments bound to the old thinking. This clearly written,well-illustrated book supplies sufficient background to beself-contained." --CHOICE
This fully revised new edition offers the most comprehensivecoverage of modern geometry currently available at an introductorylevel. The book strikes a welcome balance between academic rigorand accessibility, providing a complete and cohesive picture of thescience with an unparalleled range of topics.
Illustrating modern mathematical topics, Introduction toTopology and Geometry, Second Edition discusses introductorytopology, algebraic topology, knot theory, the geometry ofsurfaces, Riemann geometries, fundamental groups, and differentialgeometry, which opens the doors to a wealth of applications. Withits logical, yet flexible, organization, the SecondEdition:
* Explores historical notes interspersed throughout theexposition to provide readers with a feel for how the mathematicaldisciplines and theorems came into being
* Provides exercises ranging from routine to challenging,allowing readers at varying levels of study to master the conceptsand methods
* Bridges seemingly disparate topics by creating thoughtfuland logical connections
* Contains coverage on the elements of polytope theory, whichacquaints readers with an exposition of modern theory
Introduction to Topology and Geometry, Second Edition is anexcellent introductory text for topology and geometry courses atthe upper-undergraduate level. In addition, the book serves as anideal reference for professionals interested in gaining a deeperunderstanding of the topic.

Autoren/Hrsg.

Stahl, Saul

Stenson, Catherine

Inhaltsverzeichnis

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 Exercises465
References and Resources 497

Über Autor(innen)

SAUL STAHL, PhD, is Professor in the Department ofMathematics at the University of Kansas and twice the winner of theCarl B. Allendoerfer Award from the Mathematical Association ofAmerica.
CATHERINE STENSON, PhD, is Professor of Mathematics atJuniata College in Huntingdon, Pennsylvania.