Comultiplication structures on the wedge product of spheres

Homotopy multiplications on H-spaces and homotopy comultiplications on co-H-spaces play an important role in topology for many reasons. They are the duals with each other in the sense of Eckmann and Hilton, and have many kinds of examples to construct a binary operation of homotopy classes on loop multiplications and suspension comultiplications to give the group structure on the set of homotopy classes. In the current work, we investigate every possibility of homotopy comultiplications and focus on the study of associative and commutative comultiplications on the wedge product of (a sufficiently large number of) spheres as a generalization of the papers [5, 6, 19]. In particular, we observe some patterns on the types of associative and commutative comultiplications on the wedge product of spheres.