Ganea decompositions of classifying spaces
Yuri Berest, Yun Liu, Ajay C. Ramadoss
TL;DR
The paper develops Ganea-type towers for classifying spaces BG by joining fibrations F -> X -> BG and F' -> X' -> BG, producing a rationally sharp decomposition of BG under suitable hypotheses. It identifies Q_m(F,F') as free modules over H^*(BG;Q) with rank equal to dim H^*(F;Q), proves formality of the towers, and provides explicit presentations of cohomology and equivariant K-theory in key examples like flag manifolds and classifying spaces for commuting elements. The work connects topological decompositions with algebraic quasi-invariants, yielding Cohen-Macaulay structures and computable models for rational cohomology, and extends to generalized and filtered join constructions with numerous examples. An Appendix recasts the Ganea construction in the language of infinity-categories, proving convergence in hypercomplete infinity-topoi and establishing a broad conceptual framework for these decompositions.
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 Gómez, 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.