On groups of smooth maps into a simple compact Lie group, revisited

Abstract

Let $X$ be a closed smooth manifold, $G$ be a simple connected compact real Lie group, $M (G)$ be the group of all smooth maps from $X$ to $G$, and $M_0 (G)$ be its connected component for the $\mathcal C^\infty$-compact open topology. It is shown that maximal normal subgroups of $M_0 (G)$ are precisely the inverse images of the centre $Z(G)$ of $G$ by the evaluation homomorphisms $M_0 (G) \to G, \hskip.1cm \gamma \mapsto \gamma (a)$, for $a \in X$. This in turn is a consequence of a result on the group $\mathcal C^\infty_{n, G}$ of germs at the origin $O$ of $\mathbf R^n$ of smooth maps $\mathbf R^n \to G$: this group has a unique maximal normal subgroup, which is the inverse image of $Z(G)$ by the evaluation homomorphism $\mathcal C^\infty_{n, G} \to G, \hskip.1cm \underline \gamma \mapsto \underline \gamma (O)$. This article provides corrections for part of an earlier article [Harp--88].