HOMOLOGY DECOMPOSITIONS FOR p-COMPACT GROUPS

We construct a homotopy theoretic setup for homology decompositions of classifying spaces of p-compact groups. This setup is then used to obtain a subgroup decomposition for p-compact groups which generalizes the subgroup decomposition with respect to p-stubborn subgroups for a compact Lie group constructed by Jackowski, McClure and Oliver. Homology decompositions are among the most useful tools in the study of the homotopy theory of classifying spaces. Roughly speaking, a homology decomposition for a space X, with respect to some homology theory h , is a recipe for gluing together spaces, desirably of a simpler homotopy type, such that the resulting space maps into X by a map which induces an h -isomorphism. When constructing a homology decomposition for a classifying space of a group G, it is natural to do so using classifying spaces of subgroups of G. For compact Lie groups two types of mod-p homology decompositions are known: the centralizer decomposition with respect to elementary abelian p-subgroups, due to Jackowski and McClure [JM], and the subgroup decomposition with respect to certain families of p-toral subgroups, due to Jackowski, McClure and Oliver [JMO]. A p-compact group is an Fpnite loop space X (i.e., a loop space whose mod-p homology is nite), whose classifying space BX is p-complete in the sense of [BK]. These objects, de ned by Dwyer and Wilkerson [DW1], and extensively studied by them and others, are a far reaching homotopy theoretic generalization of compact Lie groups and their classifying spaces. Dwyer and Wilkerson also introduced in [DW2] a centralizer decomposition with respect to elementary abelian p-subgroups for p-compact groups, which generalizes the corresponding decomposition for compact Lie groups. The aim of this paper is to construct a subgroup decomposition for p-compact groups, analogous to the subgroup decomposition for compact Lie groups introduced by Jackowski, McClure and Oliver in [JMO]. We will in fact show that in the right setup, the Dwyer-Wilkerson theorem about existence of a centralizer decomposition for p-compact groups, with respect to their elementary abelian p-subgroups, implies the existence of subgroup decompositions with respect to certain other families of subgroups. Interestingly, as we will show, the opposite implication holds as well. More detail will be given shortly. We start by explaining some of the concepts involved. A homomorphism between pcompact groups is a pointed map : BY !BX. A subgroup of a p-compact group X is a pair (Y; ) where Y is a p-compact group and : BY !BX is a monomorphism, namely, a pointed map whose homotopy bre is Fpnite. The phrase \(Y; ) is a subgroup of X" will frequently be abbreviated by Y X. [I 1] A p-compact torus 1991 Mathematics Subject Classi cation. Primary 55R35. Secondary 55R40, 20D20.