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

Abstract

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

9 pages

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• Homotopy types of gauge groups over Riemann surfaces