Unstable splittings of classifying spaces of p-compact groups

Dwyer and Wilkerson gave a definition of a p-compact group, which is a loop space with certain properties and a good generalization of the notion of c ompact Lie groups in terms of classifying spaces and homotopy theory; e.g. every p-compact group has a maximal torus, a normalizer of the maximal toru s and a Weyl group. The belief or hope that p-compact groups enjoy most pro perties of compact Lie groups establishes a program for the classification of these objects. Following the classification of compact connected Lie gro ups, one step in this program is to show that every simply connected p-comp act group splits into a product of simply connected simple p-compact groups . The proof of this splitting theorem is based on the fact that every class ifying space of a p-compact group splits into a product if the normalizer o f the maximal torus does.