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 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