Chapter One Basic Concepts |
1 The Combinatorial Method |
2 Continuous Transformations in the Plane |
3 Compactness and Connectedness |
4 Abstract Point Set Topology |
Chapter Two Vector Fields |
5 A Link Between Analysis and Topology |
6 Sperner's Lemma and the Brouwer Fixed Point Theorem |
7 Phase Portraits and the Index Lemma |
8 Winding Numbers |
9 Isolated Critical Points |
10 The Poincaré Index Theorem |
11 Closed Integral Paths |
12 Further Results and Applications |
Chapter Three Plane Homology and Jordan Curve Theorem |
13 Polygonal Chains |
14 The Algebra of Chains on a Grating |
15 The Boundary Operator |
16 The Fundamental Lemma |
17 Alexander's Lemma |
18 Proof of the Jordan Curve Theorem |
Chapter Four Surfaces |
19 Examples of Surfaces |
20 The Combinatorial Definition of a Surface |
21 The Classification Theorem |
22 Surfaces with Boundary |
Chapter Five Homology of Complexes |
23 Complexes |
24 Homology Groups of a Complex |
25 Invariance |
26 Betti Numbers and the Euler Characteristic |
27 Map Coloring and Regular Complexes |
28 Gradient Vector Fields |
29 Integral Homology |
30 Torsion and Orientability |
31 The Poincaré Index Theorem Again |
Chapter Six Continuous Transformations |
32 Covering Spaces |
33 Simplicial Transformations |
34 Invariance Again |
35 Matrixes |
36 The Lefschetz Fixed Point Theorem |
37 Homotopy |
38 Other Homologies |
Supplement Topics in Point Set Topology |
39 Cryptomorphic Versions of Topology |
40 A Bouquet of Topological Properties |
41 Compactness Again |
42 Compact Metric Spaces |
Hints and Answers for Selected Problems |
Suggestions for Further Reading |
Bibliography |
Index |