This is a wonderfully intellectual, semi-historical approach to classical topology.



Chapter 0 gets some fundamentals out of the way. Chapter 1 is very intriguing and contains lots of ideas. First we are given a taste of the Riemann surfaces of complex analysis. These are complemented by the nonorientable surfaces, and it all leads to the classification of surfaces, which is achieved through the fundamental group and the realisations of surfaces as polygons with identifications, and this in turn leads picturesquely to covering surfaces. These simply and concisely presented ideas provide the seeds for much of the later chapters. The short chapter 2 sets up the two-way connection between topology and combinatorial group theory, which proves fruitful when the fundamental group grows into two chapters of its own (3 and 4). Then follows a sort of supplementary chapter 5 which touches on homology theory (otherwise largely neglected, but with good reason, Stillwell argues) to motivate abelianisation, which is the method we use to formally tell the fundamental groups of all surfaces apart. Chapters 2-5 were a bit slowed down by foundational issues, but now in chapters 6-8 it's all topology all the time. There are nice accounts of the classical theories of curves on surfaces (chapter 6) and knots (chapter 7). In chapter 8 we see how some of our previous ideas carry over to 3-manifolds. But ultimately 3-manifolds are deep water, with the homeomorphism problem being unsolved and all. Neither would it help to move up to 4-manifolds or higher, but at least that's not our fault so to speak because there the homeomorphism problem is in fact unsolvable. The homeomorphism problem and other fundamental problems are essentially algorithmic (i.e., given two spaces, decide whether they are different or the same) so unsolvability (non-existence of algorithms) is indeed a force to be reckoned with, so it is given its own chapter 9, naturally culminating with the unsolvability of the homeomorphism problem.



There are many ways to destroy the soul of topology. Stillwell says in the preface: "In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams." Stillwell protects us from such dangers by his emphasis on low dimensions, his insistence on the fundamental group as the best unifying tool, visualisation and illustrations, and his great respect for primary sources. The latter is reflected in frequent references and in the commented, chronological bibliography, which is very useful.