The rational LS-category of $k$-trivial fibrations

Abstract:

We provide new upper and lower bounds for the rational LS-category of a rational fibration $\xi :F\to E \to K(\mathbf {Q},2n)$ of simply connected spaces that depend on a measure of the triviality of $\xi$ which is strictly finer than the vanishing of the higher holonomy actions. In particular, we prove that if $\xi$ is $k$-trivial for some $k\ge 0$ and $H^{*}(F)$ enjoys Poincaré duality, then \begin{equation*}\operatorname {cat}_{0}E \ge \operatorname {cat}_{0}F +k.\end{equation*}