A classification of the stable type of \(BG^\wedge_p\) for compact Lie groups
Let \(p\) be a prime, and \(G\) a compact Lie group. Circumventing problems arising from the failure of the Segal conjecture for compact Lie groups, the authors study the stable homotopy type of the \(p\)-completion \(BG_p^\wedge\) of the classifying space \(BG\). An important role is played by \(p\)-toral subgroups of \(G\) (that is, by compact Lie groups which are extensions of tori by finite \(p\)-groups). It is proved (Theorem 1.1) that for two compact Lie groups \(G\) and \(G'\) the spaces \(BG_p^\wedge\) and \(B{G'}_p ^\wedge\) are stably homotopy equivalent if for every \(p\)-toral group \(P\) there is an isomorphism \(\mathbb F_p Inj(P,G)\cong \mathbb F_p Inj(P,G')\) of \(Out(P)\)-modules. Here the basis \(Inj(P,G)\) of the \(\mathbb F_p\) vector space \(\mathbb F_p Inj(P,G)\) is the set of \(G\)-conjugacy classes of continuous injections \(P\rightarrow G\). The converse of this theorem is true for finite groups (a previous paper by the authors [Topology 34, No. 3, 633-649 (1995; Zbl 0835.55011)]); it is not known to be true in general. For odd primes \(p\) and compact Lie groups \(G\) and \(G'\) the authors obtain the following characterization (Theorem 1.2): \(BG_p^\wedge\) and \(B{G'}_p^\wedge\) are stably homotopy equivalent if and only if for every \(p\)-toral group \(P\) there is a discrete approximation \(P_\infty\) [\textit{M. Feshbach}, Topology 26, 1-20 (1987; Zbl 0606.57024)] and an isomorphism \(\mathbb F_p Inj_P(P_\infty,G)\cong \mathbb F_p Inj_P(P_\infty,G')\) of \(Out(P_\infty)\) modules. Here \(Inj_P(P_\infty,G)\) is the set of \(G\)-conjugacy classes of injections \(\beta: P_\infty \rightarrow G\) such that \(\beta(P_\infty)\) is contained in a subgroup isomorphic to \(P\). In addition to this, again for odd primes, another theorem characterizes the stable homotopy type of \(BG_p^\wedge\) by finite groups. There are analogous necessary and sufficient conditions for \(BG_2^\wedge\) and \(B{G'}_2^\wedge\) having the same stable type if the generalized quaternion group of order \(16\) is not a subquotient of the Weyl group of the Sylow \(2\)-subgroup of either \(G\) or \(G'\). To prove that \(BG_p^\wedge\) and \(B{G'}_p^\wedge\) are stably equivalent, the authors show that they have the same indecomposable summands with the same multiplicities.
scientific article; zbMATH DE number 1307816