Structure theorems for actions of homeomorphism groups

Abstract: We give general classification and structure theorems for actions of groups of homeomorphisms and diffeomorphisms on manifolds, reminiscent of classical results for actions of (locally) compact groups. This gives a negative answer to Ghys' "extension problem" for diffeomorphisms of manifolds with boundary, as well as a classification of all homomorphisma $\mathrm{Homeo}_0(M) \to \mathrm{Homeo}_0(N)$ when dim(M) = dim(N) (and related results for diffeomorphisms), and a complete classification of actions of $\mathrm{Homeo}_0(S^1)$ on surfaces. This resolves many problems in a program initiated by Ghys, and gives definitive answers to conjectures of Militon and Hurtado and a question of Rubin.