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
1
12 What Is Topology Concerned With?
4
13 The Simplest Topological Invariants
8
14 The Euler Characteristic of a Graph
11
15 Intersection Index
15
16 The Jordan Curve Theorem
19
17 What Is a Curve?
22
18 Peano Curves
28
Homotopy and Homology
81
32 The Fundamental Group
83
33 Cell Decompositions and Polyhedra
87
34 Coverings
91
35 The Degree of a Mapping and the Fundamental Theorem of Algebra
95
36 Knot Groups
99
37 Cycles and Homology
104
38 Topological Products
114

Topology of Surfaces
31
22 Surfaces
33
23 The Euler Characteristic of a Surface
38
24 Classification of Closed Orientable Surfaces
42
25 Classification of Closed Nonorientable Surfaces
48
26 Vector Fields on Surfaces
55
27 The Four Color Problem
60
28 Coloring Maps on Surfaces
62
29 Wild Spheres
66
210 Knots
70
211 Linking Numbers
76
39 Fiber Bundles
117
310 Morse Theory
121
Topological Objects in Nematic Liquid Crystals VP Mineev
127
AI Nematics
128
A 3 Disclination and Topology
131
A4 Singular Points
134
A5 What Else Is There?
136
Bibliography
137
Index
139
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