On diagonal resolvable spheres

In this paper we give the complete answer to a question posed by A. Arhangelʼskii and prove that the sphere S n is diagonal resolvable if and only if S n is an H-space if and only if n ∈ { 0 , 1 , 3 , 7 } . Moreover, we prove that any upper half even dimensional Q -sphere cannot be diagonal resolvable.