STRUCTURE THEOREMS FOR ACTIONS OF HOMEOMORPHISM GROUPS

We give general classification and structure theorems for actions of groups of home-omorphisms and diffeomorphisms on manifolds, reminiscent of classical results for actions of (locally) compact groups. This gives a negative answer to Ghys's extension problem for diffeomorphisms of manifolds with boundary, as well as a classifica-tion of all homomorphisms Homeo0(M) ! Homeo0(N) when dim(M) D dim(N) (and related results for diffeomorphisms), and a complete classification of actions of Homeo0(S1) 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.