Commutativity of localized self-homotopy groups of symplectic groups
For a group-like space \(G\), and any space \(X\), the homotopy classes \([X, G]\) have a natural group structure. When \(X= G\), this is the group \([G, G]={\mathcal H}(G)\). It is not to be confused with the group of homotopy classes of self-equivalences \({\mathcal E}(G)\), where the multiplication is composition. \textit{A. Kono} and \textit{H. Ōshima} [Bull. Lond. Math. Soc. 36, No. 1, 37--52 (2004; Zbl 1041.55009)], have determined precisely when \({\mathcal H}(G)\) is commutative, in case \(G\) is a compact connected Lie group. One may go further and ask similar questions for the localizations \({\mathcal H}(G)_{(p)}\). \textit{C. A. McGibbon} [Am. J. Math. 106, 665--687 (1984; Zbl 0574.55004)] determines when \({\mathcal H}(SU(n))_{(p)}\) is homotopy commutative, and \textit{H. Hamanaka} and \textit{A. Kono} [Topology Appl. 157, No. 7, 1182--1187 (2010; Zbl 1189.55005)] determine precisely when \({\mathcal H}(SU(n))_{(p)}\) is commutative. The purpose of this paper is to answer this same question for \(({\mathcal H}(Sp(n))_{(p)}\). The method uses a technique for calculation of commutators in \([X, Sp(n)]\). It depends on \textit{T. Nagao} [J. Math. Kyoto Univ. 48, No. 1, 149--165 (2008; Zbl 1170.55003)].
scientific article; zbMATH DE number 5888205
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