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

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 ${\mathcal{C}}^{\mathrm{\infty }}$-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 $M_0(G)\to G,\gamma \mapsto \gamma (a)$, for $a\in X$. This in turn is a consequence of a result on the group ${\mathcal{C}}_{n,G}^{\mathrm{\infty }}$ 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 ${\mathcal{C}}_{n,G}^{\mathrm{\infty }}\to G,\underset{‾}{\gamma }\mapsto \underset{‾}{\gamma }(O)$. This article provides corrections for part of an earlier article [Harp–88].