Geometry & Topology

Automatic continuity for homeomorphism groups and applications

Kathryn Mann

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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.

Article information

Geom. Topol., Volume 20, Number 5 (2016), 3033-3056.

Received: 18 August 2015
Revised: 3 February 2016
Accepted: 12 March 2016
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx] 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms
Secondary: 03E15: Descriptive set theory [See also 28A05, 54H05]

homeomorphism groups automatic continuity germs of homeomorphisms


Mann, Kathryn. Automatic continuity for homeomorphism groups and applications. Geom. Topol. 20 (2016), no. 5, 3033--3056. doi:10.2140/gt.2016.20.3033.

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