This book is a comprehensive introduction to many topics in algebraic topology up to the tools currently used in research. The first 50 pages review basic topics in homology theory, homological algebra and manifolds. The text properly begins with Chapter 7 “Higher Homotopy Theory” including the Hurewicz and Whitehead theorems, and the James construction. There follow chapters on simplicial sets, fibre bundles, Hopf algebras. Chapter 11, one of the longest and most detailed, is a comprehensive treatment of spectral sequences. The text concludes with chapters on localization, generalized homology and cohomology operations.
The text was prepared in connection with a one-semester course at the Fields Institute; it was designed to bring the students, which had a year of algebraic topology, to the level where they might understand research lectures in homotopy. As the author freely admits in his preface, it is not possible to cover all these topics in detail. Those mathematicians, who take pleasure in criticizing the hard work of others, will surely ask how one can get to the Bott-Samelson theorem in 12 pages (after introductory material), Whitehead’s theorem in 18 pages, or the Hilton-Milnor theorem in 33 pages, or cover Lyusternik-Shnirel’man category in three quarters of a page. There are some legitimate questions of taste. I do not really care for the development of the Eilenberg-Moore spectral sequence which uses several spectral sequence arguments with a comparison theorem; for my taste the treatment of L. Smith (Springer Lectures Notes in Math. 134 (1969)) is much more conceptual. And the bibliography is rather short. Nevertheless, the author has pulled off a real tour de force. For a brilliant student, or just a talented student with access to a practicing topologist, this text could serve as an excellent route into some of the most exciting topics in mathematics.