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On the homotopy types of $$\mathrm{Sp}(n)$$ gauge groups. (English) Zbl 1415.55003
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 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 $$n\equiv 1 \mod 2$$.
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 $$(k,4n(2n+1))=(l,4n(n+1))$$.
Next they investigate the meaning of the number $$4n(2n+1)$$. Let $$Q_n$$ denote the quasiprojective space of rank $$n$$ defined by 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 $$x$$.
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 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 $$\epsilon =1,2$$.

##### MSC:
 55P15 Classification of homotopy type 54C35 Function spaces in general topology 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 55Q15 Whitehead products and generalizations
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