John Stillwell is a professor of mathematics at the University of San Francisco. He is also an accomplished author, having published several books with Springer, including The Four Pillars of Geometry; Elements of Algebra; Numbers and Geometry; and many more.

This is a well-balanced introduction to topology that stresses geometric aspects. Focusing on historical background and visual interpretation of results, it emphasizes spaces with few dimensions, where visualization is possible, and interaction with combinatorial group theory via the fundamental group. It also present algorithms for topological problems. Most of the results and proofs are known, but some have been simplified or placed in a new perspective.

0 Introduction and Foundations.- 0.1 The Fundamental Concepts and Problems of Topology.- 0.2 Simplicial Complexes.- 0.3 The Jordan Curve Theorem.- 0.4 Algorithms.- 0.5 Combinatorial Group Theory.- 1 Complex Analysis and Surface Topology.- 1.1 Riemann Surfaces.- 1.2 Nonorientable Surfaces.- 1.3 The Classification Theorem for Surfaces.- 1.4 Covering Surfaces.- 2 Graphs and Free Groups.- 2.1 Realization of Free Groups by Graphs.- 2.2 Realization of Subgroups.- 3 Foundations for the Fundamental Group.- 3.1 The Fundamental Group.- 3.2 The Fundamental Group of the Circle.- 3.3 Deformation Retracts.- 3.4 The Seifert-Van Kampen Theorem.- 3.5 Direct Products.- 4 Fundamental Groups of Complexes.- 4.1 Poincaré's Method for Computing Presentations.- 4.2 Examples.- 4.3 Surface Complexes and Subgroup Theorems.- 5 Homology Theory and Abelianization.- 5.1 Homology Theory.- 5.2 The Structure Theorem for Finitely Generated Abelian Groups.- 5.3 Abelianization.- 6 Curves on Surfaces.- 6.1 Dehn's Algorithm.- 6.2 Simple Curves on Surfaces.- 6.3 Simplification of Simple Curves by Homeomorphisms.- 6.4 The Mapping Class Group of the Torus.- 7 Knots and Braids.- 7.1 Dehn and Schreier's Analysis of the Torus Knot Groups.- 7.2 Cyclic Coverings.- 7.3 Braids.- 8 Three-Dimensional Manifolds.- 8.1 Open Problems in Three-Dimensional Topology.- 8.2 Polyhedral Schemata.- 8.3 Heegaard Splittings.- 8.4 Surgery.- 8.5 Branched Coverings.- 9 Unsolvable Problems.- 9.1 Computation.- 9.2 HNN Extensions.- 9.3 Unsolvable Problems in Group Theory.- 9.4 The Homeomorphism Problem.- Bibliography and Chronology.