The Centralizer Decomposition of Bg

Let G be a compact Lie group and p a fixed prime number. Recall that an elementary abelian p-group is an abelian group isomorphic to (Z/p) for some r. Jackowski and McClure showed in [10] how to decompose the classifying space BG at the prime p as a homotopy colimit of spaces of the form BCG(V ), where V is a nontrivial elementary abelian p-subgroup of G and CG(V ) is the centralizer of V in G (see §2). If the center of G is trivial then each of the centralizers CG(V ) is a proper subgroup of G, and so in this case the decomposition theorem gives an explicit way of gluing together BG, at least at p, from the classifying spaces of smaller groups. In this paper we will use this decomposition to give parallel inductive proofs of three theorems about BG; the first two theorems are already known but the third is probably new. The prime p will be fixed in everything that follows. If X is a space, let LZ/pX denote the HZ/p-localization of X constructed by Bousfield [2]. A space is said to be HZ/p-local if the natural map X −→ LZ/pX is an equivalence, or alternatively if any map f : A −→ B which induces an isomorphism on mod p homology also induces an equivalence