Evaluation maps in rational homotopy

Abstract

In the rational category of nilpotent complexes, let $E$ be an $H$-space acting on a space $X$. With mild hypotheses we show that the action on the base point $w:E\to X$ factors through a map ${\Gamma }_E:S_E\to X$, where $S_E$ is a finite product of odd-dimensional spheres and ${\Gamma }_E$ is a homotopy monomorphism. Among others, the following consequences are obtained: ${\pi }_{\ast }\left(w\right)\ne 0$ if and only if $w$ is essential and $H_{\ast }\left(w\right)\ne 0$ if and only if $X$ satisfies a strong splitting condition.