On stable summands of the classifying space of a compact lie group

Abstract

LET G be a finite group. Recent works of Benson and Feshbach [2] and Martin0 and Priddy [8,9] give a complete account of the question of stable splittings of the classifying space BG into indecomposable summands. This work is in part motivated by the work of Nishida [13] in which the notion of a dominant summand was first introduced. The purpose of this paper is to develop a framework for studying the analogous question for compact Lie groups. As in the case of finite groups where a certain generalization of the affirmative solution to Segal’s Burnside ring conjecture was used, our work begin with a theorem of May, Snaith and Zelewski [lo] which describes the set of stable maps from BQ to BK where Q is finite and K is compact Lie. If there were a similar description for Q compact Lie, one could try to imitate the work of [2] and [8] and reduce the problem to one of representation theory. However, as shown by work of Feshbach [4] and later Bauer [l] on Segal’s Burnside ring conjecture for compact Lie groups, the situation when Q is not finite is much more intricate. The key ingredient in our investigation of the stable summands of BG when G is a compact Lie group is the utilization of certain group-theoretic notions to analyze stable maps between classifying spaces of compact Lie groups. The end result is that we can define group-theoretic invariants for stable summands of BG which are invariant under stable homotopy equivalence. This puts a restriction on what kind of stable summands can occur for a given compact Lie group G. As an application, we use this framework to study two special classes of stable summands. The first is a generalized notion of a dominant summand in BG applicable for compact Lie groups. Let ( )i denote completion at the prime p in the sense of Bousfield and Kan [3]. For technical reasons, we shall add a disjoint basepoint to each classifying space of G, denoting the result by BG, . Our main theorem in this case is _ THEOREM 3.8. Suppose Xi is a dominant summand of BGi+~ where Gi is a compact Lie group with p-Sylow subgroup Ni, i = 1,2. Zf X1 is stably homotopy equivalent to X2, then N1 is isomorphic to N1.