The homotopy types of Sp(2)-gauge groups

Abstract

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