Properties of comultiplications on a wedge of spheres

In this paper we study the set of comultiplications on a wedge of a finite number of spheres. We are interested in group theoretic properties of these comultiplications such as associativity and commutativity and loop theoretic properties such as inversivity, power-associativity and the Moufang property. Our methods involve Whitehead products in wedges of spheres and the Hopf–Hilton invariants. We obtain extensive results for a restricted class of comultiplications, namely, the one-stage quadratic or cubic comultiplications.