The complete stable splitting for the classifying space of a finite group
type: Item
instance of: scholarly article
title: The complete stable splitting for the classifying space of a finite group
zbMATH Open document ID: 0752.55010
DOI: 10.1016/0040-9383(92)90067-R
author: John Martino
author: Stewart Priddy
published in: Topology
publication date: 1992-09-26T00:00:00Z
review text: This paper describes the complete stable splittings of the classifying space \(BG\) for all finite groups \(G\). It is enough to be able to compute the stable homotopy classes \(\{BG,BG\}\) for each Sylow \(p\)-subgroup \(P\) of \(G\). This is known from Carlsson's proof of the Segal Conjecture to be isomorphic to the \(p\)-adic completion of the double Burnside ring \(A(P,P)\). There is an ideal \(J_ p\subseteq \{BP,BP\}\) generated by maps of the form \(BP\to BQ\to BP\) where \(Q\subset P\) and for this ideal \(\hat Z_ p\text{Out}(P)\approx \{BP,BP\}/J_ p\). If \(X\) is a summand of \(\{BG,BG\}\) then it corresponds to a primitive idempotent \(e\in\{BP,BP\}\). If \(e\) is not in \(J_ p\) it is called a dominant summand. The multiplicity of such a summand \(X\) is equal to the dimension of \(\mathbb{F}_ p\text{Out}(P)\) over its endomorphism field. By calculating the dimension of these modules the authors obtain the result on the splitting.
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55P42
zbMATH DE Number: 54635
zbMATH Keywords: stable splittings
zbMATH Keywords: classifying space
zbMATH Keywords: stable homotopy classes
zbMATH Keywords: Segal Conjecture
zbMATH Keywords: Burnside ring
reviewed by: Jonathan Hodgson
MaRDI profile type: Publication
full work available at URL: 0040-9383(92)90067-r
OpenAlex ID: W2006728595
description: scientific article; zbMATH DE number 54635