A Combinatorial Introduction to Topology

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 Poincare 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 Poincare 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