#### Volume 20, issue 5 (2016)

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Automatic continuity for homeomorphism groups and applications

### Appendix: Frédéric Le Roux and Kathryn Mann

Geometry & Topology 20 (2016) 3033–3056
##### 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.

##### Keywords
homeomorphism groups, automatic continuity, germs of homeomorphisms
##### Mathematical Subject Classification 2010
Primary: 54H15, 57S05
Secondary: 03E15
##### Publication
Received: 18 August 2015
Revised: 3 February 2016
Accepted: 12 March 2016
Published: 7 October 2016
Proposed: Leonid Polterovich
Seconded: Danny Calegari, Bruce Kleiner
##### Authors
 Kathryn Mann Department of Mathematics University of California, Berkeley 970 Evans Hall #3840 Berkeley, CA 94720-3840 United States Frédéric Le Roux Institut de Mathématiques de Jussieu 4 place Jussieu, Case 247, 75252 Paris Cédex 5 Kathryn Mann