Stable splittings of classifying spaces of compact Lie group

Abstract

Let G be a compact Lie gorup. In this paper, we study the stable splitting of BG completed at p into a wedge sum of indecomposable spectra. When G is finite, this question has been reduced to understanding the irreducible modular representations of the outer automorphism group Out (Q) for various p-subgroups Q ⊆ G by work of [2], [14] and [18]. The principal tool used by these authors is a generalization of Segal’s Burnside ring conjecture which describes all stable maps between p-completions of the classifying spaces of p-groups. One of the problems in going from finite groups to compact Lie groups is that in the latter case one no longer has a convenient description of all stable maps between classifying spaces. Our solution to this difficulty is to pass from the ring of stable self-maps to the induced self-maps on Fp-homology. It is well-known that in terms of stable splittings, one does not lose any information by this process. There are two advantages to this approach. One is that a result of Henn [8] implies that the ring of induced self-maps on the Fp-homology of BG is finite. Two is that one can now give a more explicit description of all the induced self-maps on Fp-homology for a large class of compact Lie groups which we call p-Roquette. The latter is exactly the class of compact Lie groups for which an appropriate density theorem is valid for a generalized Segal’s Burnside ring conjecture for compact Lie groups as shown by Minami [16] whose result built upon work of Feshbach on the original Segal’s Burnside ring conjecture for compact Lie groups [7]. In particular, every compact Lie group is p-Roquette if p is odd. With these reductions, one can follow a similar procedure for splitting BG∧ p as in the case when G is finite. Recall that a compact Lie group Q is said to be p-toral if it is an extension of a torus by a finite p-group. The main result of this paper is that when G is p-Roquette, one can reduce the stable splitting of BG∧ p to the study