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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 \(\epsilon =1,2\).

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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