Volume 20, issue 5 (2016)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Automatic continuity for homeomorphism groups and applications

Kathryn Mann

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(M) to any separable group is necessarily continuous. This answers a question of C Rosendal. If N 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 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