Different constructions by Cooke, Harper and Zabrodsky and by Cohen and Neisendorfer produce torsion free finite p-local H-spaces of rank l < p − 1. The first construction goes through when l = p − 1 and we show the second does as well. However, the space produced need not be an H-space. We give a criterion for when an H-space is obtained. In the special case of rank 2 mod-3 H-spaces, we also give a practical test for when the criterion holds, and use this to give many new examples of finite H-spaces.

This paper develops the connection between the set of phantom maps from X to Y and the set of homotopy types W having the same n-type as Z, for all n, where Z = X × ΩY or Y ∨ σX. Using recent work making calculations of the space of phantom maps possible, we can give explicit constructions of various W's.

Criteria are given which characterize Co-H and Co-A maps from arbitrary double suspensions to odd dimensional spheres in terms of the maps in the EHP sequence.