Homotopy decomposition of classifying spaces via elementary Abelian subgroups
In this paper the authors show that the classifying space of any compact Lie group \(G\) can be reconstructed up to homotopy (i.e., as a homotopy colimit) from the classifying spaces of the centralizers of the nontrivial elementary abelian subgroups of \(G\). One of the steps in the argument is to show that \(H^*BG\) is the inverse limit of \(H^*BC_ G(E)\) as \(E\) ranges over the nontrivial elementary abelian subgroups of \(G\); this supplements a result of \textit{D. Quillen} [Ann. Math., II. Ser. 94, 549-572 (1971; Zbl 0247.57013)] according to which \(H^*BG\) is \(\mathcal F\)-isomorphic to the inverse limit of \(H^*BE\). The main result has been generalized to certain spaces which are not classifying spaces and proved by a different argument in \textit{W. G. Dwyer} and \textit{C. W. Wilkerson's} paper ``A cohomology decomposition theorem'' [Topology 31, No. 2, 433-443 (1992)].
classifying spaces of the centralizers of the nontrivial elementary abelian subgroups
scientific article; zbMATH DE number 54633