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.