Classifying spaces of compact Lie groups topology of p-compact groups

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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 2004 through 2005 are basically concerned with maps between classifying spaces and their applications, as well as the invariant theory. Dwyer--Wilkerson introduced the notion of 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 considered a further generalization of 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 : Suppose a-space X with an action of a finite p-group is p-complete and the cohomological dimension is finite. Then the fixed point set is an empty set if and only if the homotopy fixed point set is empty. 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. Finally, we discuss the invariant theory and the cohomology of classifying spaces. The cohomology can be expressed as an invariant ring under the action of the Weyl group. All Weyl groups are reflection groups. We obtained some results on certain reflection groups.