Automatic continuity for homeomorphism groups and applications

TL;DR

This paper proves that the group of homeomorphisms of a compact manifold has the automatic continuity property, ensuring all homomorphisms to separable groups are continuous, with applications to group structure and germs of homeomorphisms.

Contribution

It establishes the automatic continuity property for homeomorphism groups of manifolds and submanifolds, answering a longstanding question and exploring implications for group germs.

Findings

Homeo(M) has automatic continuity for compact manifolds.

Groups of germs at a point are strongly uniformly simple.

Applications include insights into the topology and structure of homeomorphism groups.

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(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, 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 Frederic Le Roux we also show, using related techniques, that the group of germs at a point of homeomorphisms of Rn is strongly uniformly simple.