Commutativity conditions for monoidal 3-categories

After recalling the various commutativity conditions which
it is natural to impose on the multiplication law in a monoidal 2-
groupoid, I will examine the corresponding question for 3-groupoids.
I will then explain the relation between this question and the occurrence
of certain 3-torsion elements in the integral homology of
the Eilenberg-Mac Lane spaces $K(A, n)$.