Universal Abelian H-spaces

The question of the existence of universal homotopy commutative and homotopy associative H-spaces (called Abelian H-spaces) is studied. Such a space T ( X ) would prolong a map from X into an Abelian H-space to a unique H-map from T into X. Examples of such pairs ( X , T ) are given and conditions are discussed which limit the possible spaces X for which such a T can exist. Contrary to published assertions, the Anick spaces are shown not to be universal Abelian H-spaces for the corresponding Moore spaces; however conditions are discussed which could lead to a universal property with respect to a more limited range of targets, and a restricted universal property is proven.