The homotopy types of Sp(2)-gauge groups

There are countably many equivalence classes of principal Sp(2) -bundles over S 4 , classified by the integer value second Chern class. We show that the corresponding gauge groups G k have the property that if there is a homotopy equivalence G k ¿G k ' , then (40,k)=(40,k ' ) , and we prove a partial converse by showing that if (40,k)=(40,k ' ) , then G k and G k ' are homotopy equivalent when localized rationally or at any prime.