Structure theorems for actions of homeomorphism groups

We give general classification and structure theorems for actions of groups homeomorphisms diffeomorphisms on manifolds, reminiscent classical results (locally) compact groups. This gives a negative answer to Ghys’s extension problem manifolds with boundary, as well all homomorphisms Homeo0(M)→Homeo0(N) when dim(M)=dim(N) (and related diffeomorphisms), complete Homeo0(S1) surfaces. resolves many problems in program initiated by Ghys, definitive answers conjectures Militon Hurtado question Rubin.