The Segal conjecture for compact Lie groups
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.
0th stable G- equivariant cohomotopy group
completion of the Burnside ring with respect to all finite subgroups
scientific article; zbMATH DE number 3980048