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 ΓE:SE→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)6=0 if and only if w is essential and H∗(w)6=0 if and only if X satisfies a strong splitting condition.
Felix, Yves and Lupton, Gregory, "Evaluation Maps in Rational Homotopy" (2007). Mathematics Faculty Publications. 188.
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