Intuitive Combinatorial Topology
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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Topology of Curves
12 What Is Topology Concerned With?
13 The Simplest Topological Invariants
14 The Euler Characteristic of a Graph
15 Intersection Index
16 The Jordan Curve Theorem
17 What Is a Curve?
18 Peano Curves
211 Linking Numbers
Homotopy and Homology
32 The Fundamental Group
33 Cell Decompositions and Polyhedra
35 The Degree of a Mapping and the Fundamental Theorem of Algebra
36 Knot Groups
37 Cycles and Homology
Topology of Surfaces
23 The Euler Characteristic of a Surface
24 Classification of Closed Orientable Surfaces
25 Classification of Closed Nonorientable Surfaces
26 Vector Fields on Surfaces
27 The Four Color Problem
28 Coloring Maps on Surfaces
29 Wild Spheres
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