Introduction to Topology and Geometry.
Title:
Introduction to Topology and Geometry.
Author:
Stahl, Saul.
ISBN:
9781118545911
Personal Author:
Edition:
2nd ed.
Physical Description:
1 online resource (533 pages)
Series:
Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts ; v.113
Pure and Applied Mathematics: A Wiley Series of Texts, Monographs and Tracts
Contents:
Cover -- Title Page -- Copyright -- Contents -- Preface -- Acknowledgments -- Chapter 1: Informal Topology -- Chapter 2: Graphs -- 2.1 Nodes and Arcs -- 2.2 Traversability -- 2.3 Colorings -- 2.4 Planarity -- 2.5 Graph Homeomorphisms -- Chapter 3: Surfaces -- 3.1 Polygonal Presentations -- 3.2 Closed Surfaces -- 3.3 Operations on Surfaces -- 3.4 Bordered Surfaces -- 3.5 Riemann Surfaces -- Chapter 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 -- Chapter 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 -- Chapter 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 -- 6.8 Polyhedral Surfaces II -- Chapter 7: Riemann Geometries -- Chapter 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 -- Chapter 9: The Fundamental Group -- 9.1 Definitions and the Punctured Plane -- 9.2 Surfaces -- 9.3 3-manifolds -- 9.4 The Poincaré Conjecture -- Chapter 10: General Topology -- 10.1 Metric and Topological Spaces -- 10.2 Continuity and Homeomorphisms -- 10.3 Connectedness -- 10.4 Compactness -- Chapter 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 -- References and Resources -- Index -- Pure and Applied Mathematics.
Abstract:
An easily accessible introduction to over three centuries of innovations in geometry Praise for the First Edition ". . . a welcome alternative to compartmentalized treatments bound to the old thinking. This clearly written, well-illustrated book supplies sufficient background to be self-contained." -CHOICE This fully revised new edition offers the most comprehensive coverage of modern geometry currently available at an introductory level. The book strikes a welcome balance between academic rigor and accessibility, providing a complete and cohesive picture of the science with an unparalleled range of topics. Illustrating modern mathematical topics, Introduction to Topology and Geometry, Second Edition discusses introductory topology, algebraic topology, knot theory, the geometry of surfaces, Riemann geometries, fundamental groups, and differential geometry, which opens the doors to a wealth of applications. With its logical, yet flexible, organization, the Second Edition: Explores historical notes interspersed throughout the exposition to provide readers with a feel for how the mathematical disciplines and theorems came into being Provides exercises ranging from routine to challenging, allowing readers at varying levels of study to master the concepts and methods Bridges seemingly disparate topics by creating thoughtful and logical connections Contains coverage on the elements of polytope theory, which acquaints readers with an exposition of modern theory Introduction to Topology and Geometry, Second Edition is an excellent introductory text for topology and geometry courses at the upper-undergraduate level. In addition, the book serves as an ideal reference for professionals interested in gaining a deeper understanding of the topic.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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