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

TL;DR

This paper characterizes the maximal normal subgroups of the group of smooth maps from a closed manifold into a simple compact Lie group, revealing their structure as inverse images of the center of the Lie group, with implications for germs of smooth maps.

Contribution

It establishes a precise description of maximal normal subgroups in the group of smooth maps into a simple Lie group, correcting previous results and extending understanding of these groups' structure.

Findings

Maximal normal subgroups are inverse images of the center of G.

Unique maximal normal subgroup in the germ group at the origin.

Provides corrections to earlier related work.

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].