The $H$-structure of low-rank torsion free $H$-spaces

Starting with a p-local space X of l odd dimensional cells, l < p-1, Cooke, Harper, and Zabrodsky constructed an H-space Y with the property that (H) over tilde (*)(Y) is generated as an exterior Hopf algebra by H-*(X). Cohen and Neisendorfer, and later Selick and Wu, reproduced this result with different constructions. We use the Selick and Wu approach to show that Y is homotopy associative and homotopy commutative if X is a suspension and l < p-2.