On the homotopy types of $\mathrm{Sp}(n)$ gauge groups
For a compact connected simple Lie group $G$ and an integer $k$, let $P_k$ be the principal $G$-bundle over $S^4$ classified by the characteristic element $k\in \mathbb{Z}= \pi_3(G)$, and let $\mathcal{G}_k(G)$ denote the gauge group of $P_k$. In this paper, the authors study the classification of homotopy types of gauge groups $\{\mathcal{G}_k(G)\}_{k\in \mathbb{Z}}$ when $G=\mathrm{Sp}(n)$. From now on, let us write $\mathcal{G}_{k,n}=\mathcal{G}_k(\mathrm{Sp}(n))$, and let $(a,b)$ denote the greatest common divisor of the integers $a$ and $b$. Note that \textit{W. A. Sutherland} [Proc. R. Soc. Edinb., Sect. A, Math. 121, No. 1--2, 185--190 (1992; Zbl 0761.55007)] previously showed that if $\mathcal{G}_{k,n}$ and $\mathcal{G}_{l,n}$ are homotopy equivalent, then $(k,n(2n+1))=(l,n(2n+1))$ for $n\equiv 0 \mod 2$ and $(k,4n(2n+1))=(l,4n(n+1))$ for $n\equiv 1 \mod 2$. \par As the first step, the authors improve the above result and they prove that if $\mathcal{G}_{k,n}$ and $\mathcal{G}_{l,n}$ are homotopy equivalent, then $(k,4n(2n+1))=(l,4n(n+1))$. \par Next they investigate the meaning of the number $4n(2n+1)$. Let $Q_n$ denote the quasiprojective space of rank $n$ defined by \textit{I. M. James} [Proc. Lond. Math. Soc. (3) 9, 115--140 (1959; Zbl 0089.39401)] and let $\iota_n:Q_n\to\mathrm{Sp}(n)$ be the natural inclusion. Then they show that the order of the Samelson product $\langle \epsilon ,\iota_n\rangle$ is $4n(2n+1)$. Remark that the order of the Samelson product $\langle \epsilon ,1_{\mathrm{Sp}(n)}\rangle$ is no less than that of $\langle \epsilon ,\iota_n\rangle$ and they try to prove that both orders are equal. Although they cannot prove this, they show that it is equal if we localize them at the latter prime $p$. More precisely, they show that $\nu_p(|\langle \epsilon ,\iota_n\rangle |)=\nu_p(4n(2n+1))$ if $(p-1)^2+1\geq 2n$, where $|x|$ is the order of the element $x$. \par As an application, they prove that $\mathcal{G}_{k,n}$ and $\mathcal{G}_{l,n}$ are $p$-local homotopy equivalent if and only if $\nu_p((k,4n(2n+1))=\nu_p((l,4n(2n+1))$ for $(p-1)^2+1\geq 2n$. Moreover, by using a result due to \textit{E. M. Friedlander} [Ann. Math. (2) 101, 510--520 (1975; Zbl 0308.55016)], they also show that $\mathcal{G}_k(\mathrm{Spin} (2n+\epsilon))$ and $\mathcal{G}_l(\mathrm{Spin} (2n+\epsilon))$ are $p$-local homotopy equivalent if and only if $\nu_p(k,4n(2n+1))=\nu_p(l,4n(2n+1))$ when $(p-1)^2+1\geq 2n \geq 6$ and $\epsilon =1,2$.
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