Automatic continuity for homeomorphism groups and applications

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 $N \subset M$ 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

References

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

Volume 20, issue 5 (2016)

Kathryn Mann

Appendix: Frédéric Le Roux and Kathryn Mann

Abstract

Keywords

Mathematical Subject Classification 2010

References

Publication

Authors