The book is divided in two halves, each of which is by one of the authors. The first one, written by W. G. Dwyer, has the title “Classifying Spaces and Homology Decompositions” and is devoted to understanding the classifying space \(BG\) of a given finite group \(G\). The second one, written by H. W. Henn, is somewhat a survey on the most recent developments on the use of Steenrod Algebra to study the mod-\(p\) cohomology \(H^{\ast }( BG) \) and is entitled “Cohomology of Groups and Unstable Modules over the Steenrod Algebra”. The reader will find in both of these two parts a good collection of exercises, examples and problems.
The classifying space \(BG\) embodies homotopically the group \(G\) in the sense that \(\pi _{1}( BG) \cong G\) and \(\pi _{k}( BG) =0\) for \(k>1\), and this is the main reason to regard the classifying space as a sort of universal object associated to a given group. The part by Dwyer starts by proving the existence of the universal space for an arbitrary discrete group, and its uniqueness up to homotopy equivalence. The first sections provide simplicial tools to construct a classifying space and give algebraic methods to compute their cohomology rings. A homology decomposition for \(BG\) is a mod \(p\) homology isomorphism \[ \text{hocolim\,}F@>\sim>p> BG \] where \(\mathbf{D}\) is a small category, \(\mathbf{Sp}\) is the category of topological spaces and continuous maps, and \(F:\mathbf{D}\rightarrow \mathbf{ Sp}\) is a functor such that for each \(d\in Ob(\mathbf{D})\) the space \(F( d) \) is weakly equivalent to the classifying space \( BH_{d}\) for some \(H_{d}<G\). The search for a good homology decomposition goes through a process of stricking a balance among the complexity of the small category \(\mathbf{D}\) and the functor \(F\), starting from a collection \(\mathcal{C}\) of subgroups of \(G\), closed under conjugation. Dwyer’s part of the book is concerned with various techniques to reach this goal.
Quillen’s theorem describes the cohomology of a compact Lie group in terms of elementary abelian groups. Now since the collection of abelian groups is amply regarded as a category, the homotopy type of \(BG\) is determined by the diagram of elementary abelian subgroups of \(G\). By some reasons related to the properties of a particular homology decomposition of \(BG\), its cohomology ring \(H^{\ast }( BG) \) rests onthe action of the isotropy groups on the cohomology of elementary abelian subgroups. Furthermore, the Steenrod squares generate the so called Steenrod algebra which acts on the cohomology ring, and provides a very useful interpretation of Quillen’s theorem, as a statement on \(H^{\ast }( BG;\mathbb{F} _{p}) \), considered as an object in the category \(\mathcal{U}\)of unstable modules over the Steenrod algebra. The Adem relations are proved to be crucial in establishing some properties of the category \(\mathcal{U}\), particularly useful to build the tools needed to study the action of the Steenrod algebra. This is the subject of the second part of the book, where the interested reader will also find an illuminating proof of Quillen’s theorem as well as Lannes’ generalization of it.
Both parts of this book show a very carefully chosen route, so it minimizes the reader’s effort. They provide useful introductions to several topics and techniques that otherwise would be very difficult to find in a single piece of literature, so this book will be of plenty of interest to both topologists and algebraists.