Stable homotopy classification of \(BG_ p^ \land\)
type: Item
instance of: scholarly article
title: Stable homotopy classification of \(BG_ p^ \land\)
zbMATH Open document ID: 0835.55011
DOI: 10.1016/0040-9383(94)00040-R
author: John Martino
author: Stewart Priddy
published in: Topology
publication date: 1995-10-22T00:00:00Z
review text: The authors present proofs of their results announced in an earlier note [Bull. Am. Math. Soc., New Ser. 27, No. 1, 165-170 (1992; Zbl 0754.55015)]. The main theorem is that, given a prime number \(p\), for two finite groups \(G\) and \(G'\) the \(p\)-completions of their classifying spaces \(BG\) and \(BG'\) (or, equivalently, the \(p\)- localizations of \(BG\) and \(BG'\) have the same stable homotopy type if and only if any one of the following two equivalent conditions holds: (1) For every \(p\)-group \(Q\), the \(\mathbb{F}_p\)-vector spaces \(\mathbb{F}_p \text{Rep} (Q,G)\) and \(\mathbb{F}_p \text{Rep} (Q,G')\) are isomorphic as \(\text{Out} (Q)\)-modules; here the basis \(\text{Rep} (Q,G)\) of \(\mathbb{F}_p \text{Rep} (Q,G)\) consists of \(G\)-conjugacy classes of homomorphisms from \(Q\) into \(G\). (2) For every \(p\)-group \(Q\), \(\mathbb{F}_p \text{Inj} (Q,G)\) and \(\mathbb{F}_p \text{Inj} (Q,G')\) are isomorphic as \(\text{Out} (Q)\)- modules; here \(\text{Inj} (Q,G) \subseteq \text{Rep} (Q, G)\) consists of conjugacy classes of injective homomorphisms. In the proof, the authors essentially use their results obtained in a previous paper [Topology 31, No. 1, 143-156 (1992; Zbl 0752.55010)]. The concept of linkage allows them to use representation theory in describing the stable summands of the \(p\)-localization of \(BG\); the multiplicity of a stable summand turns out to be the rank of a certain matrix. This matrix then allows to determine the stable homotopy type of the \(p\)- localization of \(BG\). Several interesting applications of the main theorem, primarily to the case when \(G\) and \(G'\) are finite groups with normal Sylow \(p\)- subgroups, are also proved in this paper.
reviewed by: Július Korbaš
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55P42
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 794443
zbMATH Keywords: completions
zbMATH Keywords: localization
zbMATH Keywords: representation spaces
zbMATH Keywords: finite groups
zbMATH Keywords: classifying spaces
zbMATH Keywords: stable homotopy type
zbMATH Keywords: \(p\)-group
zbMATH Keywords: linkage
zbMATH Keywords: finite groups with normal Sylow \(p\)-subgroups
MaRDI profile type: Publication
Recommended article: A classification of the stable type of 𝐵𝐺
Recommended article: Alternative stable homotopy classification of \(BG_{p}^{\land}\)
Recommended article: Unstable homotopy classification of
Recommended article: RETRACTIVE TRANSFERS AND p-LOCAL FINITE GROUPS
Recommended article: Homotopy equivalences of \(p\)-completed classifying spaces of finite groups
Recommended article: A classification of the stable type of \(BG^\wedge_p\) for compact Lie groups
Recommended article: Q4656983
Recommended article: The homotopy theory of fusion systems
Recommended article: Q4010211
Recommended article: Stable splitting and cohomology of p-local finite groups over the extraspecial p-group of order p^3 and exponent p
description: scientific article; zbMATH DE number 794443