Homotoy Theory of Classifying Spaces

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The research on the classifying spaces of compact Lie groups has been one of the major area in Homotoy Theory. Our results obtained during 1997 through 1998 are basically concerned with maps between classifying spaces and their applications. Dwyer-Wilkerson defined ap-compact group and studied its properties. The purely homotopy theoritic 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 wrtten by Moeller in the AMS Bulletin summarizes their work. Further progress on the homotopy theory of the classifying spaces of p-compact groups are being made.We state here our main results. First, we consider the maps of p-compact groups of the form BX * BY*BZ.The main theorem shows that if the restriction map on BY is a weak epimorphism, then the restriction on BX should factor through the classifying spaces of the center of the p-compact group Z.Next, for G =S^3 * .. * S^3, let X be a genus of BG.We investigate the monoid of rational equivalences of X, denoted by epsilon(X). It is shown that a submonoid of epsilon_0(X), denoted by delta_00(X), determines the decomposability of the space X.We also show converses to some known results for the classifying spaces of p-toral groups or p-compact toral group. Suppose G is a compact Lie group. The following results are obtained. If there is a positive integer k such that the n-th homotopy groups of the p-completion of BG are zero for all n <greater than or equal> k then the loop space of this space is a p-compact toral group. If the canonical map Rep(G, K)*[BG, BK] is bijective for any compact connected Lie group K, then G is a p-toral group. in addition, our work containesa research on the conditions of a compact Lie group that its loop space of the p-completed classifying space be a p-compact group.