On the homotopy types of $\mathrm{Sp}(n)$ gauge groups

Abstract

Let ${\mathcal{G}}_{k,n}$ be the gauge group of the principal $Sp\left(n\right)$–bundle over $S^4$ corresponding to $k\in \mathbb{Z}\cong {\pi }_3\left(Sp\left(n\right)\right)$. We refine the result of Sutherland on the homotopy types of ${\mathcal{G}}_{k,n}$ and relate it to the order of a certain Samelson product in $Sp\left(n\right)$. Then we classify the $p$–local homotopy types of ${\mathcal{G}}_{k,n}$ for ${\left(p-1\right)}^2+1\ge 2n$.