Classifying spaces of Kač-Moody groups
Kač-Moody groups are essentially obtained by integrating certain in general infinite dimensional Lie algebras. The representation theory of these Lie algebras is a priori quite closely related to the representation theory of finite dimensional Lie algebras, and the paper under review arose from asking whether these similarities are reflected in the homotopy theoretic properties of Kač-Moody groups. It turned out to be useful to investigate this question in a more general context. Let \(p\) be a fixed prime. Starting from the category \({\mathcal L}\) of compact Lie groups, the authors consider a new class of topological groups \(K_1({\mathcal L})\). By definition a compactly generated Hausdorff topological group \(G\) is contained in \(K_1({\mathcal L})\) if there is a finite \(G\)-\(CW\)-complex \(X\) whose isotropy groups are in \({\mathcal L}\) and for which for any finite \(p\)-group \(\pi \subset G\) the fixed point space \(X^\pi\) is \(p\)-acyclic. The definition is a variation of a corresponding notion introduced by \textit{P. H. Kropholler} and \textit{G. Mislin} in [Comment. Math. Helv. 73, 122-136 (1998; Zbl 0927.20033)], where these two authors considered discrete groups \(G\) rather than topological groups, and acyclic fixed point sets instead of \(p\)-acyclic fixed point sets. Kač-Moody groups are contained in \(K_1({\mathcal L})\), however the class \(K_1({\mathcal L})\) contains way more groups. For example it also contains all parabolic subgroups of Kač-Moody groups, and if \(G\in K_1({\mathcal L})\) then also all centralizers \(C_G(\pi)\) for finite \(p\)-subgroups \(\pi \subset G\) are in \(K_1({\mathcal L})\). The authors now show that indeed many results familiar within the context of compact Lie groups also hold or have analog versions for topological groups \(G\in K_1({\mathcal L})\). For example they show that the mod-\(p\)-cohomology algebra of \(G\) is noetherian. They also show that Lannes' \(T\)-functor computes the mod-\(p\) cohomology of the classifying spaces of the centralizers of elementary abelian \(p\)-subgroups, and that the spaces of maps from classifying spaces of finite \(p\)-groups to \(BG\) can be described in terms of classifing spaces of centralizers up to \(p\)-completion. Moreover they obtain an analog of the centralizer decomposition theorem of Jackowski and McClure which computes \(BG\) through a colimit of classifying spaces of centralizers of elementary abelian \(p\)-subgroups of \(G\) up to \(p\)-completion.
centralizer decomposition of classifying space
scientific article; zbMATH DE number 1807899