Elementary Topology : A Combinatorial and Algebraic Approach.
- 1st ed.
- 1 online resource (237 pages)
Front Cover -- Elementary Topology: A Combinatorial and Algebraic Approach -- Copyright Page -- Table of Contents -- PREFACE TO THE REVISED EDITION -- PREFACE -- CHAPTER 1. SOME EXAMPLES OF SURFACES -- 1.1 Coordinates on a Sphere and Torus -- 1.2 The Topological Sphere and Torus -- 1.3 Properties of the Sphere and Torus -- 1.4 The Cylinder and Möbius Band -- 1.5 Additional Representations of the Möbius Band -- 1.6 The Projective Plane -- EXERCISES -- CHAPTER 2. THE CLASSIFICATION OF SURFACES -- 2.1 Surfaces and Their Equations -- 2.2 Combinatorial Equivalence -- 2.3 The Canonical Equation -- 2.4 Combinatorial Invariants of a Surface -- 2.5 Topological Surfaces -- EXERCISES -- CHAPTER 3. COMPLEX CONICS AND COVERING SURFACES -- 3.1 Complex Conies -- 3.2 Covering Surfaces -- 3.3 Some Additional Examples of Riemann Surfaces -- EXERCISES -- CHAPTER 4. MAPPINGS INTO THE SPHERE -- 4.1 Winding Number of a Plane Curve -- 4.2 Mappings into the Plane -- 4.3 The Brouwer Degree -- 4.4 Applications of the Winding Number in Complex Analysis -- EXERCISES -- CHAPTER 5. VECTOR FIELDS -- 5.1 Vector Fields on the Plane -- 5.2 A Geographical Application -- 5.3 Vector Fields and Hydrodynamics -- 5.4 Vector Fields and Differential Equations -- 5.5 Vector Fields on a Sphere -- 5.6 Mappings of a Sphere into Itself -- EXERCISES -- CHAPTER 6. NETWORK TOPOLOGY -- 6.1 Introduction -- 6.2 Boundary and Coboundary -- 6.3 Paths, Circuits, and Trees -- 6.4 Basic Circuits -- 6.5 The Kirch hoff-Maxwell Laws -- 6.6 A Transportation Problem -- EXERCISES -- CHAPTER 7. SOME THREE-DIMENSIONAL TOPOLOGY -- 7.1 Three-Dimensional Manifolds -- 7.2 Orientability -- 7.3 Manifolds of Configurations -- 7.4 Topological Products and Fiber Bundles -- EXERCISES -- BIBLIOGRAPHY -- INDEX.