Minami-Webb splittings and some exotic \(p\)-local finite groups
type: Item
instance of: scholarly article
title: Minami-Webb splittings and some exotic \(p\)-local finite groups
zbMATH Open document ID: 1113.55012
author: Assaf Libman
published in: Mathematische Zeitschrift
publication date: 2007-03-13T00:00:00Z
review text: The goal of ``homology decomposition'' is to present, up to \(p\)-completion in the sense of \textit{A. K. Bousfield} and \textit{D. M. Kan} [Homotopy Limits, Completions and Localizations. Lectures Notes in Mathematics 304. Springer (1972; Zbl 0259.55004)], a classifying space of a group by means of the classifying spaces of its proper subgroups. The main papers in this subject are \textit{S. Jackowski, J. McClure} and \textit{B. Oliver}'s work in [Topology 31, No. 1, 113--132 (1992; Zbl 0754.55014) and Ann. Math. (2) 135, No. 1, 183--226 (1992; Zbl 0758.55004), ibid. No. 2, 227--270 (1992; Zbl 0771.55003)] and \textit{W. G. Dwyer}'s achievements [Topology 36, No. 4, 783--804 (1997; Zbl 0872.55014), Sharp homology decomposition for classifying spaces of finite groups. Group representations: cohomology, group actions and topology (Seattle, WA, 1996), Proc. Symp. Pure Math. 63, 197--220 (1998; Zbl 0891.55021)]. The introduction of \(p\)-local finite groups by \textit{C. Broto, R. Levi} and \textit{B. Oliver} in [J. Am. Math. Soc. 16, No. 4, 779--856 (2003; Zbl 1033.55010)] re-opened the subject. At present the goal is to decompose the classifying space of a \(p\)-local finite group by means of classifying spaces of finite groups [see e.g. Broto, Levi and Oliver's paper and a paper (to appear) of \textit{A. Libman}]. Every finite group \(G\) gives rise to a \(p\)-local finite group \((S,\mathcal{F},\mathcal{L})\), where \(S\) is a Sylow \(p\)-subgroup of \(G\). A \(p\)-local finite group is called ``exotic'' if its underlying fusion system \(\mathcal{F}\) cannot be associated with a finite group. The classifying space of a \(p\)-local finite group \((S,\mathcal{F},\mathcal{L})\) is by definition the \(p\)-completed nerve \({|\mathcal{L}|}_{p}\) of the linking system \(\mathcal{L}\). In the non-exotic case it is homotopy equivalent to \(BG_{p}^{\wedge}\). A consequence of \textit{N. Minami}'s theorem 6.5 in [Trans. Am. Math. Soc. 351, No. 11, 4481--4513 (1999; Zbl 0931…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55N91
Mathematics Subject Classification ID: 20D15
zbMATH DE Number: 5133227
zbMATH Keywords: homology decomposition
zbMATH Keywords: \(p\)-local finite groups
MaRDI profile type: Publication
full work available at URL: s00209-006-0030-2
OpenAlex ID: W2081080159
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DOI: 10.1007/S00209-006-0030-2
description: scientific article; zbMATH DE number 5133227