Automatic continuity for homeomorphism groups and applications

Abstract

Let $M$ be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of $M$ has the automatic continuity property: any homomorphism from $Homeo\left(M\right)$ to any separable group is necessarily continuous. This answers a question of C Rosendal. If $N\subset M$ is a submanifold, the group of homeomorphisms of $M$ that preserve $N$ also has this property.

Various applications of automatic continuity are discussed, including applications to the topology and structure of groups of germs of homeomorphisms. In an appendix with Frédéric Le Roux we also show, using related techniques, that the group of germs at a point of homeomorphisms of ${ℝ}^n$ is strongly uniformly simple.