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
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
38 Topological Products
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
211 Linking Numbers
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1-cycles abelian Betti numbers called cell decomposition chromatic number circle closed surface complement space compute connected graph Consider continuous deformation contours of type curvilinear image defined degree diametrically opposite points direction disclination disk edge path embedded endpoints equation Euler characteristic Example exterior fiber bundle finite number follows function fundamental group glue gluing graph G handles holes homologous to zero homology groups homotopic integral cycle interior intersection index isotopic Jordan curve theorem knot Let Q linking number midline minimum point Mobius strip modulo multivalued function number of edges number of points number of vertices obtain a surface one-dimensional parallel Peano curve point XQ polyhedra polyhedron Problems projective plane prove 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 three-space topological invariant topological product torus traversed vector field vertex yields zero-dimensional