Automatic continuity for homeomorphism groups of noncompact manifolds

Abstract

We extend the proof of automatic continuity for homeomorphism groups of manifolds to non-compact manifolds and manifolds with marked points and their mapping class groups. Specifically, we show that, for any manifold $M$ homeomorphic to the interior of a compact manifold, and a set $X\subset M$ homeomorphic to the union of a Cantor set and finite set, the relative homeomorphism group $\mathrm{Homeo}(M,X)$ and the mapping class group $\mathrm{Homeo}(M,X)/{\mathrm{Homeo}}_0(M,X)$ have the property that any homomorphism from such a group to any separable topological group is necessarily continuous.

References in corpus (1)

• The first integral cohomology of pure mapping class groups

Cited by in corpus (6)

• Big mapping class groups with hyperbolic actions: classification and applications
• Mapping class groups with the Rokhlin property
• Self-similar surfaces: involutions and perfection
• Three perfect mapping class groups
• Mapping class groups of surfaces with noncompact boundary components
• A class of open surfaces with algorithmically solvable homeomorphism problem