The Segal conjecture for compact Lie groups
type: Item
instance of: scholarly article
title: The Segal conjecture for compact Lie groups
zbMATH Open document ID: 0606.57024
DOI: 10.1016/0040-9383(87)90016-4
author: Mark Feshbach
published in: Topology
publication date: 1987-01-01T00:00:00Z
review text: G. Carlsson has proved the Segal conjecture for finite groups: if \(G\) is a finite group, then the Segal map \(\pi_{G^*}(S^ 0)^{{\hat{\;}}}\to \pi_{S^*}(BG^+)\) is an isomorphism, where \(\pi_{G^*}(S^ 0)^{{\hat{\;}}}\) denotes \(\pi_{G^*}(S^ 0)\) completed at the augmentation ideal I(G) in the Burnside ring \(A(G)\). The Burnside ring of a compact Lie group has been defined by tom Dieck, who has shown with Petrie that this ring is isomorphic to the 0th stable \(G\)-equivariant cohomotopy group of \(S^ 0\). The author points out that the naive generalization of the Segal conjecture fails to hold. In case \(G\) is a compact Lie group, a different completion is required, as \(G=O(2)\) shows (Part B of paper). However, the naive generalization holds if \(G=S^ 1\). The subtler completion of the Burnside ring with respect to all finite subgroups of \(G\) is required. Let \(C(G)\) be the completion of \(I(G)\) under the coarsest topology which makes the maps \(I(G)\to I(F)\) continuous, where \(F\) runs through all finite subgroups of \(G\) and \(I(F)\) is given the \(l\)-adic topology. This paper shows that the Segal map \(I(G)\to \pi_ S^ 0(BG)\) induces an isomorphism under completion \(C(G)\approx \pi_ S^ 0(BG)\) at all odd primes. It is not known if it holds at \(p=2\), but the author shows that it holds here under the additional hypotheses that rank \(G\leq 8\) or that the Weyl group of \(G\) does not contain any subquotient isomorphic to a generalized quaternion group of order 16.
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 55Q91
Mathematics Subject Classification ID: 55P60
Mathematics Subject Classification ID: 55Q10
zbMATH DE Number: 3980048
zbMATH Keywords: Segal conjecture
zbMATH Keywords: Burnside ring of a compact Lie group
zbMATH Keywords: 0th stable G- equivariant cohomotopy group
zbMATH Keywords: completion of the Burnside ring with respect to all finite subgroups
zbMATH Keywords: Weyl group
Wikidata QID: Q115364135
reviewed by: Arunas Liulevicius
MaRDI profile type: Publication
full work available at URL: 0040-9383(87)90016-4
OpenAlex ID: W1994066712
description: scientific article; zbMATH DE number 3980048