Evaluation maps in rational homotopy

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 → X factors through a map Γ E : S E → X , where S E is a finite product of odd-dimensional spheres and Γ E 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 X satisfies a strong splitting condition.