This book was written as a text for a one-year (English) under-graduate topology course, but, as the author points out, there is plenty of scope for selecting from it for shorter courses or courses at a lower level. The author had two aims in writing the book. ``Firstly, to make sure the student sees a variety of different techniques and applications involving point set, geometric, and algebraic topology, without delving too deeply into any particular area. Secondly, to develop the reader's geometric insight; topology is after all a branch of geometry.'' The reviewer feels the book achieves these aims. Moreover, the material is well motivated throughout, and the introductory chapter takes the form of a self-contained motivational essay. The techniques which are introduced are complemented by solid applications and are sensibly chosen for the technique rather than only the application. For instance, in Chapter 7 surfaces are classified using surgery, rather than by the classical combinatorial proof, which, though probably faster, gives less geometric insight and is more a means to an end than an expandable technique. (That surgery can be done is proved via a triangulation of the surface, the proof of the existence of which is naturally only sketched.) Other topics covered are: basic point set topology (two of ten chapters); identification spaces, including orbit spaces of topological group actions; the fundamental group; simplicial homology of finite complexes (up to invariance of homology); degree of maps of spheres; Lefschetz fixed point theorem; covering spaces; knots, including Seifert surfaces, the infinite cyclic cover of the knot exterior, and the Alexander polynomial. The problems which follow each section are well chosen, including many designed to increase the reader's geometric insight. A book at this level cannot touch on all possible topics, and probably most readers will find in any such book some topic or technique which they feel has come short. For instance, differential topology is never mentioned in this book, but it could well be supplied from J. Milnor's book [Topology from a differentiable viewpoint, Univ. Virginia, Charlottesville, Va., 1965; MR0226651] which complements this book quite nicely. In fact, as is evident from the above list of topics, most of the book is based on, but by no means rigidly tied to, the simplicial approach. The book is definitely ``pure'' in the sense that applications outside the field are seldom, if ever, mentioned. However, one of the most useful things a future nontopologist can take away from a topology course, to use when he needs topology elsewhere, is a good geometric intuition, and this book should certainly supply it.
Reviewer: Neumann, W. D. [form MathSciNet]