Intuitive Combinatorial Topology

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Springer Science & Business Media, Mar 30, 2001 - Mathematics - 141 pages
Topology is a relatively young and very important branch of mathematics. It studies properties of objects that are preserved by deformations, twistings, and stretchings, but not tearing. This book deals with the topology of curves and surfaces as well as with the fundamental concepts of homotopy and homology, and does this in a lively and well-motivated way. There is hardly an area of mathematics that does not make use of topological results and concepts. The importance of topological methods for different areas of physics is also beyond doubt. They are used in field theory and general relativity, in the physics of low temperatures, and in modern quantum theory. The book is well suited not only as preparation for students who plan to take a course in algebraic topology but also for advanced undergraduates or beginning graduates interested in finding out what topology is all about. The book has more than 200 problems, many examples, and over 200 illustrations.
 

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Contents

Topology of Curves
3
12 What Is Topology Concerned With?
6
13 The Simplest Topological Invariants
10
14 The Euler Characteristic of a Graph
13
15 Intersection Index
17
16 The Jordan Curve Theorem
21
17 What Is a Curve?
24
18 Peano Curves
30
211 Linking Numbers
78
Homotopy and Homology
83
32 The Fundamental Group
85
33 Cell Decompositions and Polyhedra
89
34 Coverings
93
35 The Degree of a Mapping and the Fundamental Theorem of Algebra
97
36 Knot Groups
101
37 Cycles and Homology
106

Topology of Surfaces
33
22 Surfaces
35
23 The Euler Characteristic of a Surface
40
24 Classification of Closed Orientable Surfaces
44
25 Classification of Closed Nonorientable Surfaces
50
26 Vector Fields on Surfaces
57
27 The Four Color Problem
62
28 Coloring Maps on Surfaces
64
29 Wild Spheres
68
210 Knots
72
38 Topological Products
116
39 Fiber Bundles
119
310 Morse Theory
123
Topological Objects in Nematic Liquid Crystals VP Mineev
129
AI Nematics
130
A 3 Disclination and Topology
133
A4 Singular Points
136
Bibliography
139
Index
141
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Page xi - In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.

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