$p$-compact groups as subgroups of maximal rank of Kac-Moody groups

Abstract

In [28], Kitchloo constructed a map f : BX → BK∧ p where K is a certain KacMoody group of rank two, X is a rank two mod p finite loop space and f is such that it induces an isomorphism between even dimensional mod p cohomology groups. Here B denotes the classifying space functor and (−)p denotes the Bousfield-Kan Fp-completion functor ([8]). This space X —or rather the triple (X∧ p , BX ∧ p , e) where e : X ' ΩBX— is a particular example of what is known as a p-compact group. These objects were introduced by Dwyer and Wilkerson in [15] as the homotopy theoretical framework to study finite loop spaces and compact Lie groups from a homotopy point of view. The foundational paper [15] together with its many sequels by Dwyer-Wilkerson and other authors represent now an active, well established research area which contains some of the most important recent advances in homotopy theory. While p-compact groups are nowadays reasonably well understood objects, our understanding of Kac-Moody groups and their classifying spaces from a homotopy point of view is far from satisfactory. The work of Kitchloo in [28] started a project which has also involved Broto, Saumell, Ruiz and the present author and has produced a series of results ([2], [3], [10]) which show interesting similarities between this theory and the theory of p-compact groups, as well as non trivial challenging differences. The goal of this paper is to extend the construction of Kitchloo that we have recalled above to produce rank-preserving maps BX → BK∧ p for a wide family of p-compact groups X. These maps can be understood as the homotopy analogues to monomorphisms, in a sense that will be made precise in section 13. We prove: