Constructing modular classifying spaces
The paper addresses a classical question of Steenrod: ``What spaces have polynomial algebras as mod p cohomology?''. For odd p, \textit{W. G. Dwyer}, \textit{H. R. Miller} and \textit{C. Wilkerson} [Homotopical uniqueness of classifying spaces (unpublished)] showed that a realizable polynomial algebra should be isomorphic to the ring of invariants of a reflection group in \(GL_ n({\mathbb{F}}_ p)\) and moreover this group should lift to \(GL_ n({\mathbb{Z}}^{\hat {\;}}_ p)\). Such a group must be a product of groups from the Shephard-Todd list [\textit{G. C. Shephard} and \textit{J. A. Todd}, Can. J. Math. 6, 274-304 (1954; Zbl 0055.143)]. For some groups from the list and some primes the author constructs spaces whose \({\mathbb{F}}_ p\)-cohomology is the corresponding group of invariants (in some cases the ring is not known to be polynomial). The spaces are obtained as homotopy colimits of certain diagrams indexed over carefully chosen small categories. To calculute cohomology of the homotopy colimit the author uses the Bousfield-Kan spectral sequence [\textit{A. K. Bousfield} and \textit{D. M. Kan}, Homotopy limits, completion and localization, Lect. Notes Math. 304 (1972; Zbl 0259.55004), Ch. XII. 5.8]. He shows that all higher inverse limits which appear in the \(E_ 2\)-term vanish. The diagrams and calculations are particularly simple because the groups chosen from the Shephard-Todd list have Sylow p- subgroups of prime order. Besides new examples, the paper provides another construction of some complicated spaces developed by Zabrodsky as well as spaces whose cohomology is isomorphic to those of classifying spaces of exceptional Lie groups. The author rightly expected that the idea, which he attributes to Jean Lannes, of constructing spaces with prescribed cohomology as homotopy colimits of spaces with well-understood cohomology will have further applications. Recently, \textit{W. G. Dwyer} and \textit{C. W. Wilkerson} [A new finite loop space at the prime two (unpublished)] used it to construct an \({\mathbb{F}}_ 2\)-complete space BD(4) whose \({\mathbb{F}}_ 2\)- cohomology is isomorphic as an algebra over the Steenrod algebra to \(H^*(B({\mathbb{Z}}/2)^ 4;{\mathbb{F}})^{GL(4,{\mathbb{F}}_ 2)}\).
ring of invariants of a reflection group in \(GL_ n({\mathbb{F}}_ p)\)
classifying spaces of exceptional Lie groups
scientific article; zbMATH DE number 4143027
scientific article