This carefully written text presents a new approach to a first course in algebraic topology. The main novelty is defining the homology groups of a (pointed path-connected) CW complex as the homotopy groups of its infinite symmetric product SP X. Cohomology groups are defined as homotopy classes of maps into Eilenberg-MacLane spaces, which are defined as infinite symmetric products of Moore spaces.

This differs dramatically from most standard texts by making no mention of singular or simplicial homology. Cellular homology is discussed, in order to allow calculations and to approach the usual Kunneth and Universal Coefficient theorems.

It is somewhat closer to the 1975 texts of Gray and Switzer, but very different from them, as well. Both of them introduced homology and cohomology using spectra, and defined Eilenberg-MacLane spaces by attaching cells to kill unwanted homotopy classes.

The Dold-Thom theorem that underlies the symmetric product approach to homology states that SP X → SP(X/A) is a quasifibration. A thorough proof of this is given in an appendix, apparently the first proof in English of this fundamental result. By the way, this book is a translated and expanded version of the authors’ original Spanish text, but shows no signs of being a translation. The English is excellent, and the misprints are few and minor.

The first third of the book is fairly standard, dealing with the fundamental group, homotopy groups, cofibrations and fibrations, and covering spaces, in this order. There is a thorough discussion of the duality between cofibrations and fibrations. Covering spaces are treated as a special kind of fibration, and their relationship with π_{1}(−) is not emphasized as much as in most books. Much of the theory of covering spaces is relegated to exercises. Throughout, exercises are embedded in the text; there are no sets of exercises at the end of chapters. The purpose of exercises in this book seems to be primarily for exposition of material.

One nice topic here, not found in most texts, is a classifying space for n-fold covering spaces, namely F_{n}(R^{∞})/Σ_{n}. This is illustrative of the sophisticated level of this text. In comparison, the associative property of π_{1}(−) is established by just a formula for the homotopy between (ω_{1}ω_{2})ω_{3} and ω_{1}(ω_{2}ω_{3}), without the usual picture as an aid.

The novel approach to homology used in this book can be disconcerting to one who is used to the standard treatment. One must wait a long time (or know to skip over a lot of material) before getting to the homology groups of familiar spaces such as surfaces and real projective spaces. Nowhere does this book mention that the first homology group is the abelianization of the fundamental group. No explanation is made for why the ubiquitous sign in the anticommutativity of the cup product is present.

Another big omission compared to standard texts is the treatment of Kunneth and Universal Coefficient theorems. Here the reader is directed to other texts. Statements of results are given, but Tor and Ext are not even defined. This book never even tells what a chain complex is or how its homology groups are defined. The reader is referred to a text such as MacLane’s for such essentials.

After the chapters on homology and cohomology are four thorough chapters on vector bundles, K-theory, Adams operations, and characteristic classes. A very nice and thorough proof of the Adams-Atiyah proof of nonexistence of division algebras is given. In these chapters, the text again expects a lot of the reader. In the first section (9 pages), where a fairly standard definition of vector bundle is given, there are no examples. In the next section, it is shown how a map from X into the space Pr(V) of all projections (idempotent endomorphisms) of a vector space V gives rise to a vector bundle. Some of the standard examples are then given from this sophisticated perspective.

The final chapter deals with spectra and the Brown Representability Theorem. It treats the distinction between prespectra and spectra as in May’s “Concise” 1999 text.

In addition to the appendix on the Dold-Thom theorem, there is one with a thorough topological proof of the complex Bott Periodicity Theorem, following a 1999 paper of the first and third authors. This utilizes the quasifibration theme of the book.

In conclusion, this text has developed a truly new approach to introductory algebraic topology. The introduction explains that this approach is in line with the recent work of Voevodsky. But a reader with no background in homological algebra will need to refer to a supplementary text.