Torsion in finite H-spaces and the homotopy of the three-sphere

Let X be a 2-connected p-local finite H-space with a single cell in dimension three. We give a simple cohomological criterion which distinguishes when the inclusion i: S3i⟶X has the property that the loop of its three-connected cover is null homotopic. In particular, such a null homotopy implies that πm(i)=0 for m≥4. Applications are made to Harper's rank 2 finite H-space and simple, simply-connected, compact Lie groups.