Homotopy equivalences of \(p\)-completed classifying spaces of finite groups
type: Item
instance of: scholarly article
title: Homotopy equivalences of \(p\)-completed classifying spaces of finite groups
zbMATH Open document ID: 1042.55008
author: Carles Broto
author: Ran Levi
author: Bob Oliver
published in: Inventiones Mathematicae
publication date: 2003-08-17T00:00:00Z
review text: The paper deals with the homotopy properties of the \(p\)-completed classifying space \(BG^\wedge_p\) of a finite group \(G\) for a given prime \(p\). More precisely the following two questions are studied: when are two \(p\)-completed classifying spaces \(BG^\wedge_p\) and \(BG^{\prime\wedge}_p\) for finite groups \(G\) and \(G'\) homotopy equivalent, and what is the homotopy type of the topological monoid \(\Aut(BG^\wedge_p)\) of homotopy self equivalences of \(BG^\wedge_p\). Concerning the first question, \textit{J. Martino} and \textit{S. Priddy} stated in [Math. Proc. Camb. Philos. Soc. 119, 119--137 (1996; Zbl 0855.55008)] that \(BG^\wedge_p\) and \(BG^{\prime\wedge}_p\) are homotopy equivalent if and only if there is a fusion preserving isomorphism \(S \to S'\) between the \(p\)-Sylow subgroups of \(G\) and \(G'\), where an isomorphism \(\varphi:S \to S'\) is called fusion preserving if for any two subgroups \(P\) and \(Q\) of \(S\) a group isomorphism \(\alpha:P \to Q\) is induced by conjugation in \(G\) if and only if the corresponding isomorphism \(\varphi\alpha\varphi^{-1}:\varphi(P) \to \varphi(Q)\) is induced by a conjugation in \(G'\). The proof Martino and Priddy gave in [loc. cit.] however turned out to be incorrect, as found out by the authors of the paper under consideration. Instead of fusion preserving isomorphisms the authors investigate linking categories. The linking category \({\mathcal L}_p(G)\) of a group \(G\) is a special case of a ``categorie associée à une \(p\)-localité'' in the sense of \textit{L. Puig} [Structure locale dans les groupes finis, Bull. Soc. Math. Fr., Suppl., Mém. 47 (1976; Zbl 0355.20024)]. The objects of \({\mathcal L}_p(G)\) are the \(p\)-subgroups of \(G\), and for two \(p\)-subgroups \(P\) and \(Q\) the set of morphisms \(\text{Mor}_{{\mathcal L}_p(G)}(P,Q)\) is given by the quotient of \(N_G(P,Q)=\{x\in G\mid xPx^{-1} \leq Q\}\) by the action of \(O^p(C_G(P))\), where \(O^p(C_G(P))\) denotes the smallest normal su…
reviewed by: Michael Joachim
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55P15
Mathematics Subject Classification ID: 55Q52
Mathematics Subject Classification ID: 55R37
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 1965474
zbMATH Keywords: classifying spaces of groups
zbMATH Keywords: maps between classifying spaces
zbMATH Keywords: fusion categories
zbMATH Keywords: linking categories
MaRDI profile type: Publication
full work available at URL: s00222-002-0264-5
OpenAlex ID: W2003095911
DOI: 10.1007/S00222-002-0264-5
description: scientific article; zbMATH DE number 1965474