The homotopy types of $Sp(n)$-gauge groups over $S^{4m}$

Abstract: Let $m$ and $n$ be two positive integers such that $m < n$. Denote by $P_{n,k}$ the principal $Sp(n)$-bundle over $S^{4m}$ and $\mathcal{G}{k,m}(Sp(n))$ be the gauge group of $P{n,k}$ classified by $k\varepsilon'$, where $\varepsilon'$ is a generator of $\pi_{4m}(B(Sp(n)))\cong\mathbb{Z}$. In this article, we will partially classify the homotopy types of $\mathcal{G}{k,m}(Sp(n))$ by giving a lower bound for the number of homotopy types of $\mathcal{G}{k,m}(Sp(n))$. Also, in special cases $Sp(3)$-gauge groups over $S^8$ and $Sp(4)$-gauge groups over $S^{12}$ we give an upper bound for the number of homotopy types of these gauge groups.