Homotopy theory splits into two branches: Stable and unstable homotopy theory. For stable homotopy we have several monographs and textbooks: From Adams’ masterpiece Stable Homotopy and Generalised Homology (Chicago University Press, 1974) to Ravenel’s Complex Cobordism and Stable Homotopy Groups of Spheres (Second Edition, AMS-Chelsea, 2004). But for unstable homotopy theory, textbooks or monographs have been rather scarce. Thus the publication of the book under review fills a large gap in the literature, collecting in one place most of the techniques and results for the calculation of (unstable) homotopy groups of spaces.
The main objective of this book is to introduce the reader to some recent developments in unstable homotopy theory, focusing on those aspects that are needed to present the most recent results on the exponents of homotopy groups, proved by the author, F. Cohen, J. Moore and P. Selick. A simply connected space X has exponent pk at the prime p, if pk annihilates the p-primary torsion of the homotopy groups of X. For some important spaces, such as spheres Sn or spaces with few cells (Moore and Peterson spaces, for example) some exponents have been obtained in the last decades.
Most of these results could be traced back to some theorems that Serre proved in the 1950s, for example that all simply connected complexes with finitely many cells in each dimension have finitely generated homotopy groups, or that odd-dimensional spheres have only one non finite cyclic homotopy group, in the dimension of the sphere. Of course there were important results before Serre’s, such as Hopf’s theorem that the third homotopy group of the 2-sphere S2 is isomorphic to the infinite cyclic group Z, but Serre was the first to obtain global results and the methods that he used were also important: spectral sequences and localization away from finite sets of primes.
For the exponents of the homotopy groups of spheres, the timeline is as follows: in the second half of the 1950s, I. M. James showed that 22n annihilates the 2-primary component of the homotopy groups of the sphere S2n+1. H. Toda generalized the invariants introduced by James to prove the odd-primary analog, i.e., that for an odd prime p, p2n annihilates the p-primary component of the homotopy groups of S2n+1. There was no further progress on this topic until P. Selick proved in his Princeton thesis in 1977 that for p an odd prime, p annihilates the p-primary component of the homotopy groups of the 3-sphere S3. Selick’s theorem was generalized by F. Cohen, J. Moore and J. Neisendorfer who proved in 1979–1981 that if p is an odd prime, then pn annihilates the p-primary components of the homotopy groups of S2n+1. All these results were shown to be the best possible by B. Gray. It is worth noticing that for the prime 2 it is known that James theorem is not the best possible, but the best bound has not been found yet.
With the aim of giving a complete and self-contained exposition of these important results, including the methods and techniques that are used, the author has collected in one volume many important topics whose importance goes beyond the stated goal. Hence, we find in the first chapter a systematic exposition of homotopy groups with coefficients in an abelian group, showing for example that they satisfy a universal coefficient sequence and are functorial for the space and for the coefficient group, thus they also give rise to a Bockstein long exact sequence and Bockstein spectral sequences. For the special case when the coefficient group is a finite cyclic group Z/kZ, there is a mod-k Hurewicz homomorphism from the homotopy groups with coefficients in Z/kZ to the homology of the space with the same coefficients. Most of the techniques used to construct these homotopy groups are the now familiar fibration and cofibration sequences with Moore or Peterson spaces as inputs.
The second main topic is localization and completion. Serre’s theorems and several of the subsequent results on exponents for the homotopy groups of spheres were proved inverting some or all primes to localize the corresponding homotopy or homology groups. Chapter two gives a general set-up for all localizations in algebraic topology using the Bousfield and Dror Farjoun theory of localization, which inverts specific maps between spaces, showing when necessary that the general theory recovers the classical one.
Using the tools of localization and homotopy groups with coefficients, the author introduces several important invariants in homotopy theory: Hilton-Hopf, James-Hopf and Toda-Hopf invariants, and more importantly, Samelson products. Details of the construction and main properties of these invariants are in chapters three to six.
The fourth main topic, (graded) Lie algebras and universal enveloping algebras, is studied in chapters eight and nine. Together with the homological algebra of chapter 10, all these results are put to work in chapter eleven to prove the results on exponents that we recalled at the beginning of this review. Actually, if the reader is willing to check the results quoted from previous chapters or is more or less familiar with them, chapter 11 is almost self-contained.
This is a beautiful subject, topology at its best, with algebra playing along point-counterpoint, if I am allowed to use a musical analogy. The author has gone a long way to make the book self-contained. All results are given complete proofs, with only elementary facts from algebra or algebraic topology assumed. The book is well-organized, with topics developed in full and linked with each other.
I would say that this book belongs to the bookshelf of anyone interested in homotopy theory. Moreover, it could be used as a text for an intermediate or advanced course in algebraic topology or for self-study, since it comes equipped with exercises in all sections.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.