Ganea decompositions of classifying spaces

Abstract

We study homotopy decompositions of the classifying spaces $BG$ of compact connected Lie groups obtained by (relative) fiber-cofiber construction. Given a pair of Borel fibrations $ F \to E \to BG $ and $F' \to E' \to BG $, this construction yields a tower (telescope) of spaces $ X_{m}(F,F') $ over $BG$ indexed by $ \mathbb{Z}_+ $ that converges in the sense that $\text{hocolim} \,(X_{m})\,$ is weakly homotopy equivalent to $BG$. We determine cohomological conditions on the fibrations that produce the spaces $X_{m}(F,F')$ with properties similar to those of the spaces of quasi-invariants of Weyl groups constructed by the first and third authors. We prove that, under these conditions, the resulting homotopy decompositions of $BG$ are sharp (over $\mathbb{Q}$), the spaces $X_{m}(F,F')$ are rationally formal and Cohen-Macaulay, their cohomology rings being finite rank free modules over $H^*(BG, \mathbb{Q})$. We construct many examples which include the fundamental (maximal torus) fibration $ G/T \to BT \to BG $ as well as the universal fibration $\, E_{\rm com}G_{\bf 1} \to B_{\rm com}G_{\bf 1} \to BG \,$ for the classifying space $B_{\rm com}G$ of commuting elements in $G$ introduced by Adem and G\'{o}mez, as the first fibration in the pair. In most cases, we give an explicit presentation for the (equivariant) cohomology rings in terms of characteristic classes and compute the (equivariant) $K$-theory of the spaces involved. The paper contains an Appendix, where we re-examine the topological fiber-cofiber construction in an abstract setting, proving an $\infty$-categorical extension of the classical Ganea Theorem.