Comultiplications on a wedge of two spheres (Q547327)

The first author of this paper showed in [Co-H-spaces. James, I. M. (ed.), Handbook of algebraic topology. Amsterdam: North-Holland. 1143--1173 (1995; Zbl 0941.55001)] that the comultiplication of the \(n\)-sphere with \(n\geq2\) is unique and commutative up to homotopy. On the other hand, for a wedge of two spheres, it is known that there are many comultiplications with many different properties. In this paper, the authors show that the number of comultiplications on a wedge of two spheres is the product of the order of the homotopy groups of certain spheres. The authors also study the conditions for each comultiplication to be homotopy associative or homotopy commutative.