The homotopy types of Sp(2)-gauge groups

Abstract

There are countably many equivalence classes of principal Sp(2)-bundles over S4, classified by the integer value second Chern class. We show that the corresponding gauge groups Gk have the property that if there is a homotopy equivalence Gk≃Gk′, then (40,k)=(40,k′), and we prove a partial converse by showing that if (40,k)=(40,k′), then Gk and Gk′ are homotopy equivalent when localized rationally or at any prime.