The homotopy types of Sp(3)-gauge groups

Let Gk denote the gauge group of the principal Sp(3)-bundle over S4 with first symplectic Pontryagin class k∈H4S4=Z. We show that when localised at an odd prime there is a homotopy equivalence Gk≃Gl if and only if (21,k)=(21,l). We also give bounds on the number of 2-local homotopy types amongst the Gk. Our results follow from a thorough examination of the commutator map c:Sp(1)∧Sp(3)→Sp(3) and we determine its order to be either 4⋅3⋅7=84, 8⋅3⋅7=168 or 16⋅3⋅7=336. In particular its order at odd primes is completely determined.