Odd primary homotopy types of $Spin(n)$-gauge groups over $S^8$

Abstract

Let $G$ be one of $Sp(3)$, $Spin(7)$ or $Spin(8)$. Also, let $P_k$ be the principal $G$-bundle over $S^8$ and $\mathcal{G}_k(G)$ be the gauge group of $P_k$ classified by $k\varepsilon$, where $\varepsilon$ a generator of $\pi_8(B(G))\cong\mathbb{Z}$. In this article, localized at an odd prime $p$, we partially classify the homotopy types of $\mathcal{G}_k(G)$.