Homotopy commutativity in \(p\)-localized gauge groups (Q2866551)

Let \(G\) be a simple, simply connected, compact Lie group and let \(P\to S^4\) be a principal \(G\)-bundle, classified by the second Chern class, \(c_{2}(P)\in H^4(BG)\cong \mathbb{Z}\). If \(c_{2}(P)=k\), we denote by \(\mathcal{G}_{k}(G)\) the gauge group of \(G\). In the paper under review, the authors study how the homotopy commutativity of \(G\), or its failure, determines that of \(\mathcal{G}_{k}(G)\).NEWLINENEWLINERecall that \(G\) is rationally homotopy equivalent to a product of spheres, \(G\simeq_{\mathbb{Q}}\prod_{i=1}^{\ell} S^{2n_{i}-1}\). In [Am. J. Math. 106, 665--687 (1984; Zbl 0574.55004)], \textit{C. A. McGibbon} showed that the usual loop multiplication on \(G\) is homotopy commutative when localized at \(p\) in precisely the following cases: \(p>2n_{\ell}\); \(G=Sp(2)\) and \(p=3\); \(G=G_{2}\) and \(p=5\).NEWLINENEWLINEAfter a localisation at \(p\), as in McGibbon's result, the authors prove that the gauge group, \(\mathcal{G}_{k}(G)\), is homotopy commutative if \(p>2n_{\ell}+1\). They obtain also results for certain matrix groups. For instance, \(\mathcal{G}(SU(n))\) is not homotopy commutative at \(p\) if \(p<2n\), except possibly, for \(p=2\) and \(n\in\{2,3,4\}\), or \(p=3\), \(n\in\{2,3\}\) and \((3,k)=1\). Moreover, if \(p=2n+1\), then \(\mathcal{G}_{k}(SU(n))\) is homotopy commutative at \(p\) if \(k\equiv 0\) mod \(p\) and, not homotopy commutative at \(p\) if \(k\not\equiv 0\) mod \(p\). The previous remaining cases are still open. Other results concern the groups \(Sp(n)\) and \(Spin(n)\).