Automatic Continuity for Homeomorphism Groups and Big Mapping Class Groups

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(M,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(M,X)/Homeo0(M,X). In other words, the algebraic structure of these groups is extremely rigid and determines their topology in a very strong way.