Intuitive Combinatorial TopologyTopology 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. |
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 |
| 139 | |
| 141 | |
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Common terms and phrases
1-cycles a₁ abelian Algebra b₁ Betti numbers C₁ called cell decomposition chromatic number circle closed surface complement space connected graph Consider continuous deformation curvilinear image defined diametrically opposite points direction disclination disk embedded endpoints equation Euler characteristic Example exterior fiber bundle field d(r finite number follows function fundamental group G₁ glue gluing graph G h₁ handles holes homeomorphic homology groups homotopic interior intersection index Jordan curve theorem knot l₁ Let Q linking number M₁ mapping f midline minimum point Möbius strip modulo number of edges number of points number of vertices obtain a surface one-dimensional parallel Peano curve point xo polygonal trail polyhedra polyhedron Problems projective plane r₁ regions saddle point segment self-intersections Show shown in Figure simple closed curve singular points spanning tree sphere square stationary points surface homeomorphic surface Q surface with boundary tangent topological invariant topological product torus traversed V₁ v₁(x vector field vertex zero-dimensional
Popular passages
Page xi - In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain.