Automatic Continuity for Homeomorphism Groups and Big Mapping Class Groups

Abstract

We show that for any manifold M, compact or homeomorphic to the interior of a compact manifold, the homeomorphism group of M has automatic continuity. The same is true for the relative homeomorphism group $Homeo(\mathit{M},\mathit{X})$ where X is homeomorphic to the union of a Cantor set and a (possibly, empty) finite set, and the big mapping class group $Homeo(\mathit{M},\mathit{X})/{Homeo}_0(\mathit{M},\mathit{X})$. In other words, the algebraic structure of these groups is extremely rigid and determines their topology in a very strong way.