The homotopy types of $\operatorname{Sp}(2)$-gauge groups

Abstract

There are countably many equivalence classes of principal $\operatorname{Sp}(2)$-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.