Preface vii
Notation xi

Basic Definitions 1

Chapter 1. Graphs 5
1. Topological and Geometric Properties of Graphs 5
2. Homotopy Properties of Graphs 29
3. Graph Invariants 47

Chapter 2. Topology in Euclidean Space 55
1. Topology of Subsets of Euclidean Space 55
2. Curves in the Plane 63
3. The Brouwer Fixed Point Theorem and Sperner’s Lemma 72

Chapter 3. Topological Spaces 87
1. Elements of General Topology 87
2. Simplicial Complexes 99
3. CW-Complexes 117
4. Constructions 130

Chapter 4. Two-Dimensional Surfaces, Coverings, Bundles, and
Homotopy Groups 139
1. Two-Dimensional Surfaces 139
2. Coverings 149
v
vi Contents
3. Graphs on Surfaces and Deleted Products of Graphs 157
4. Fibrations and Homotopy Groups 161

Chapter 5. Manifolds 181
1. Definition and Basic Properties 181
2. Tangent Spaces 199
3. Embeddings and Immersions 207
4. The Degree of a Map 220
5. Morse Theory 239

Chapter 6. Fundamental Groups 257
1. CW-Complexes 257
2. The Seifert–van Kampen Theorem 266
3. Fundamental Groups of Complements of Algebraic Curves 279

Hints and Solutions 291
Bibliography 317
Index 325