Evaluation maps in rational homotopy

In the rational category of nilpotent complexes, let EE be an HH-space acting on a space XX. With mild hypotheses we show that the action on the base point w:E→X factors through a map ΓE:SE→XΓE:SE→X, where SESE is a finite product of odd-dimensional spheres and ΓEΓE is a homotopy monomorphism. Among others, the following consequences are obtained: π∗(w)≠0 if and only if w is essential and H∗(w)≠0 if and only if XX satisfies a strong splitting condition.