Stable splittings of classifying spaces of finite groups
type: Item
instance of: scholarly article
title: Stable splittings of classifying spaces of finite groups
zbMATH Open document ID: 0752.55008
DOI: 10.1016/0040-9383(92)90068-S
published in: Topology
publication date: 1992-09-26T00:00:00Z
review text: This paper describes the splitting of the stable homotopy of classifying spaces of finite groups. From Carlsson's proof of the Segal Conjecture one knows that the stable homotopy classes \(\{BG';BG\}\) are isomorphic to the completion of a Burnside group \(A(G',G)\). \(A(G',G)\) is generated as a free abelian group by the basis elements \(\zeta_{H',\phi}\) where \(H'\) is a subgroup of \(G'\) and \(\phi\) is a homomorphism \(\phi: H'\to G\). The correspondence \(X\mapsto | X^ H|\) defines a map \(A(G',G)\to \mathbb{Z}\). Putting these two together the authors define a module \(M(G',G)\) generated as an abelian group by maps \(f_{H',\psi}\). By calculating the formula for the action of \(A(P,P)\) on the coadjoint module \(M(P,P)\) the authors obtain the splitting results.
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 54636
zbMATH Keywords: splitting
zbMATH Keywords: stable homotopy
zbMATH Keywords: classifying spaces
zbMATH Keywords: finite groups
zbMATH Keywords: Segal Conjecture
zbMATH Keywords: Burnside group
author: David Benson
author: Mark Feshbach
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
Wikidata QID: Q105419679
description: scientific article; zbMATH DE number 54636