Homotopy Methods in a Generalization of Lie Group Theory

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The research on the classifying spaces of compact Lie groups is one of the major area in Homotopy Theory. Our results obtained during 2002 through 2003 are basically concerned with maps between classifying spaces and their applications Dwyer-Wilkerson defined a p-compact group and studied its properties. The purely homotopy theoretic object appears to be a good generalization of a compact Lie group. A p-compact group has rich structure, such as a maximal torus, a Weyl group, etc. A note written by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of p-compact groups are being made. We state here our main results. First, we generalize a result of Dror Farjoun and Zabrodsky on the relationship between fixed point sets and homotopy fixed point sets, which is related to the generalized Sullivan Conjecture : For a finite p-group Π, suppose a Π -space X is F_p-complete and cd_p(X) is finite. Then X^πis an empty set if and only if the homotopy fixed point set X^<hπ> is empty. As an application, we discuss extension problems considering actions on homogeneous spaces of p-compact groups. Next, we consider a problem on the conditions of a compact Lie group G that its loop space of the p-completed classifying space be a p-compact group for a set of primes. In particular, we discuss the classifying spaces BG that are p-compact for all primes when the groups are certain subgroups of simple Lie groups. A necessary and sufficient condition that BG be p-compact toral for all primes has been obtained. We ask if BH is p-compact for a set of primes when H is a subgroup of a simple Lie group G and obtaine certain results.