Elementary Topology (eBook)

A Combinatorial and Algebraic Approach

Donald W. Blackett (Autor)

eBook Download: PDF

2014 | 1. Auflage
236 Seiten
Elsevier Science (Verlag)
978-1-4832-6253-6 (ISBN)

Infos

Elementary Topology: A Combinatorial and Algebraic Approach focuses on the application of algebraic methods to topological concepts and theorems. The publication first elaborates on some examples of surfaces and their classifications. Discussions focus on combinatorial invariants of a surface, combinatorial equivalence, surfaces and their equations, topological surfaces, coordinates on a sphere and torus, and properties of the sphere and torus. The text then examines complex conics and covering surfaces and mappings into the sphere, including applications of the winding number in complex analysis, mappings into the plane, winding number of a plane curve, covering surfaces, and complex conies. The book examines vector fields, network topology, and three-dimensional topology. Topics include topological products and fiber bundles, manifolds of configurations, paths, circuits, and trees, vector fields and hydrodynamics, vector fields on a sphere, and vector fields and differential equations. The publication is highly recommended for sophomores, juniors, and seniors who have completed a year of calculus.

Inhalt

Front Cover 1
Elementary Topology: A Combinatorial and Algebraic Approach 4
Copyright Page 5
Table of Contents 8
PREFACE TO THE REVISED EDITION 6
PREFACE 6
CHAPTER 1. SOME EXAMPLES OF SURFACES 14
1.1 Coordinates on a Sphere and Torus 14
1.2 The Topological Sphere and Torus 25
1.3 Properties of the Sphere and Torus 31
1.4 The Cylinder and Möbius Band 41
1.5 Additional Representations of the Möbius Band 48
1.6 The Projective Plane 53
EXERCISES 60
CHAPTER 2. THE CLASSIFICATION OF SURFACES 66
2.1 Surfaces and Their Equations 66
2.2 Combinatorial Equivalence 74
2.3 The Canonical Equation 79
2.4 Combinatorial Invariants of a Surface 88
2.5 Topological Surfaces 92
EXERCISES 94
CHAPTER 3. COMPLEX CONICS AND COVERING SURFACES 98
3.1 Complex Conies 98
3.2 Covering Surfaces 107
3.3 Some Additional Examples of Riemann Surfaces 116
EXERCISES 119
CHAPTER 4. MAPPINGS INTO THE SPHERE 123
4.1 Winding Number of a Plane Curve 123
4.2 Mappings into the Plane 129
4.3 The Brouwer Degree 133
4.4 Applications of the Winding Number in Complex Analysis 137
EXERCISES 142
CHAPTER 5. VECTOR FIELDS 146
5.1 Vector Fields on the Plane 146
5.2 A Geographical Application 151
5.3 Vector Fields and Hydrodynamics 155
5.4 Vector Fields and Differential Equations 157
5.5 Vector Fields on a Sphere 160
5.6 Mappings of a Sphere into Itself 164
EXERCISES 167
CHAPTER 6. NETWORK TOPOLOGY 171
6.1 Introduction 171
6.2 Boundary and Coboundary 175
6.3 Paths, Circuits, and Trees 178
6.4 Basic Circuits 179
6.5 The Kirch hoff-Maxwell Laws 183
6.6 A Transportation Problem 186
EXERCISES 197
CHAPTER 7. SOME THREE-DIMENSIONAL TOPOLOGY 202
7.1 Three-Dimensional Manifolds 202
7.2 Orientability 207
7.3 Manifolds of Configurations 214
7.4 Topological Products and Fiber Bundles 226
EXERCISES 229
BIBLIOGRAPHY 232
INDEX 234

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 1-4832-6253-7 / 1483262537
• ISBN-13: 978-1-4832-6253-6 / 9781483262536