CONNECTED FINITE LOOP-SPACES WITH MAXIMAL TORI

Finite loop spaces are a generalization of compact Lie groups. However , they do not enjoy all of the nice properties of compact Lie groups. For example, having a maximal torus is a quite distinguished property. Actually, an old conjecture, due to Wilkerson, says that every connec ted finite loop space with a maximal torus is equivalent to a compact connected Lie group. We give some more evidence for this conjecture by showing that the associated action of the Weyl group on the maximal t orus always represents the Weyl group as a crystallographic group. We also develop the notion of normalizers of maximal tori for connected f inite loop spaces, and prove for a large class of connected finite loo p spaces that a connected finite loop space with maximal torus is equi valent to a compact connected Lie group if it has the right normalizer of the maximal torus. Actually, in the cases under consideration the information about the Weyl group is sufficient to give the answer. All this is done by first studying the analogous local problems.