A co-H-space is a space X together with a map \(X\to X\vee X\), called the comultiplication, whose composition with each projection \(X\vee X\to X\) is homotopic to the identity map. If X is a rational space then X is called a rational co-H-space. It is well known that a rational co-H- space, X, has the homotopy type of a wedge of spheres. The latter space admits a standard comultiplication arising from the pinching map. However, the co-H-space X need not be co-H-equivalent to a wedge of spheres. The paper begins by showing the equivalence of comultiplications with certain Lie algebra homomorphisms, via Quillen minimal models. This gives necessary and sufficient conditions for a rational co-H-space to admit infinitely many homotopy classes of comultiplications (resp. of homotopy non-associative comultiplications). Sections 4 and 5 contain a study of two further topics concerning comultiplications on a rational co-H-space: homotopy commutativity and homotopy inverses.
Most of the previous results carry over to wedges of ordinary spheres, but other require modifications. Examples illustrate how certain phenomena, when considered over \({\mathbb{Z}}\), are more complicated than when considered over \({\mathbb{Q}}\).