Constructing modular classifying spaces
type: Item
instance of: scholarly article
title: Constructing modular classifying spaces
zbMATH Open document ID: 0697.55002
DOI: 10.1007/BF02765884
author: Jaume Aguadé
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: 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p, \textit{W. G. Dwyer}, \textit{H. R. Miller} and \textit{C. Wilkerson} [Homotopical uniqueness of classifying spaces (unpublished)] showed that a realizable polynomial algebra should be isomorphic to the ring of invariants of a reflection group in \(GL_ n({\mathbb{F}}_ p)\) and moreover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 tw…
reviewed by: Stefan Jackowski
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 55T99
Mathematics Subject Classification ID: 55U99
zbMATH DE Number: 4143027
zbMATH Keywords: polynomial algebras as mod p cohomology
zbMATH Keywords: ring of invariants of a reflection group in \(GL_ n({\mathbb{F}}_ p)\)
zbMATH Keywords: Shephard-Todd list
zbMATH Keywords: homotopy colimits
zbMATH Keywords: Bousfield-Kan spectral sequence
zbMATH Keywords: classifying spaces of exceptional Lie groups
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4143027