On the I(G)-adic topology of the Burnside ring of compact Lie groups
Let G be a compact Lie group and A(G) the Burnside ring of compact G-ENRs [\textit{T. tom Dieck}, Math. Ann. 215, 235-250 (1975; Zbl 0313.57030)]. For subgroups \(H\subset G\) there are homomorphisms: \(Res^ G_ H: A(G)\to A(H)\) and \(Ind^ G_ H: A(H)\to A(G)\). The augmentation ideal I(G) in A(G) is the kernel of the restriction to the trivial group. In this paper the structure of \(A(G)^{\wedge}\), the I(G)-adic completion of A(G), is described in terms of p-adic completions of the augmentation ideals of certain subgroups of G. Let T be a maximal torus of G and for each prime p dividing \(| NT/T|\) pick a p-Sylow subgroup \(F_ p\subset NT/T\). In the following statement \(N_ p\subset NT\) is the preimage of \(F_ p\) and \((M)_ p^{\wedge}\) denotes the p-adic completion of a module M. Theorem: As topological modules, \(A(G)^{\wedge}\cong {\mathbb{Z}}\oplus I(G)^{\wedge}\) and \(I(G)^{\wedge}\cong \Sigma^{\oplus}(Ind^ G_{N_ p}I(N_ p))_ p^{\wedge}\). An explicit basis is given for \(Ind^ G_{N_ p}I(N_ p)\), which is finitely generated if and only if the action of \(F_ p\) on T is trivial. One corollary of this result is that \(A(G)^{\wedge}\) and \(\pi^ 0_ s(BG)\) are different for G a nonabelian, connected Lie group. This shows that the I(G)-adic topology is not appropriate for generalizing the Segal conjecture from finite groups to compact Lie groups [\textit{G. Carlsson}, Ann. Math., II. Ser. 120, 189-224 (1984; Zbl 0586.55008)]. Note that when G is finite the above result reduces the study of \(A(G)^{\wedge}\) to Burnside rings of p-groups. The method of proof for the main theorem is to construct continuous homomorphisms \(Ind^{\wedge}: \Sigma^{\oplus}A(N_ p)^{\wedge}\to A(G)^{\wedge}\) and \(Res^{\wedge}: A(G)^{\wedge}\to \Sigma^{\oplus}A(N_ p)^{\wedge}\) from the induction and restriction maps above. The composite \(Ind^{\wedge}\circ Res^{\wedge}\) is invertible and on \(I(N_ p)\) the p-adic and \(I(N_ p)\)-adic topologies coincide. From this the splitting of \(I(G)^{\wedge}\) follows directly.
stable equivariant cohomology groups of spheres
scientific article; zbMATH DE number 3975905