For the topologist the algebraically defined cohomology groups \(H^*(\pi,A)\), at least when the coefficient ring \(A\) has trivial module structure over the finite group \(\pi\), are to be thought of as the cohomology groups of an Eilenberg-Maclane space \(K(\pi,1)\). The ‘internal’ method of their calculation depends on knowing the answer for groups of prime power order, the ‘external’ method on restriction from some larger group, for example a symmetric or finite linear group. In recent years considerable progress has been made in the understanding of how these groups are built up, for example, in principle it is possible to calculate \(H^*(\pi,\mathbb{Z})\) when the Sylow subgroups \(\pi_p\) are of order \(p^t\) with \(t<4\) for all primes dividing the order. Furthermore, at least for the subring generated by Chern classes, it is possible to give an upper bound to the exponent in cohomology, which in general lies strictly between the exponent of the group and its order. Note that in terms of Postnikov systems there are no other invariants of \(K(\pi,1)\) to be investigated. The author of the monograph under review avoids this apparent paucity by replacing the classifying space \(B\pi=K(\pi,1)\) with its \(p\)-completion, albeit under the restriction that \(\pi\) is \(p\)-perfect (i.e. generated by commutators and elements of \(p\)-power order). He justifies this by pointing out firstly that, for \(X=B\pi\) the completion \(X^\wedge\) is homotopy equivalent to the product of the \(X^\wedge_p\), and that secondly, \(p\)-perfection is adequate up to taking covering spaces. Furthermore in contrast to the original space \(B\pi\) its \(p\)-localizations have interesting Postnikov towers. The main results which he proves are:
1. If \(p^t\) is the highest power of \(p\) dividing the order of the finite \(p\)-perfect group \(\pi\) then \(p^t\) annihilates the reduced homology of \(\Omega B\pi^\wedge_p\) with \(\mathbb{Z}_{(p)}\) coefficients. There is a similar result for stable homotopy. This is achieved using a specific model for the chain algebra, which the author claims may have other applications.
2. The Postnikov tower of \(\Omega B\pi^\wedge_p\) has infinitely many non-vanishing \(k\)-invariants. In particular, \(\pi_*B\pi^\wedge_p\) is non-trivial in arbitrarily high dimensions.
Let \(S^n\{p^t\}\) be the homotopy fibre of a degree \(p^t\)-self-map of the \(n\)-sphere \(S^n\), and define a space to be of type \(S^n\{p^t\}\) if it has the same \(\mathbb{F}_p\)-cohomology as an algebra over \(A_p\) and has the same Bockstein operators.
3. For certain groups \(\pi\) it is possible to describe the homotopy type of \(\Omega B\pi^\wedge_p\) in terms of these spaces \(S^n\{p^t\}\), for example this holds for finite special linear and symplectic groups, and in the case \(p=2\) for groups having dihedral or semi-dihedral 2-Sylow subgroups. These results depend heavily on knowledge of the ordinary cohomology of the groups concerned, and were originally designed as evidence in favour of conjectured nice ‘spherical resolutions’ for an arbitrary group \(\pi\). Unfortunately, as the author himself admits, the original conjecture has turned out to be false, and at the end of the final section he is left floundering, trying to find a replacement.
Many of the results in this monograph are local versions of results originally proved globally, i.e. for perfect rather than \(p\)-perfect groups etc., by F. Cohen. The various chapters are attractively written, but at the end this reviewer was forced to ask himself ‘What is this all for?’. At present it seems that these results are of interest only to professional homotopy theorists, and that, until further notice, it is the cohomology of \(B\pi\) itself, rather than these variants which will be of interest to mathematicians at large.