On the reconstruction problem for factorizable homeomorphism groups and foliated manifolds

For a group G of homeomorphisms of a regular topological space X and a subset U ⊆ X , set G : = { g ∈ G | g ↾ ( X ∖ U ) = Id } . We say that G is a factorizable group of homeomorphisms, if for every open cover U of X, ⋃ U ∈ U G generates G. m I H be factorizable groups of homeomorphisms of X and Y respectively, and suppose that G, H do not have fixed points. Let φ be an isomorphism between G and H. Then there is a homeomorphism τ between X and Y such that φ ( g ) = τ ○ g ○ τ − 1 for every g ∈ G . m A strengthens known theorems in which the existence of τ is concluded from the assumption of factorizability and some additional assumptions. m II = 1 , 2 let ( X ℓ , Φ ℓ ) be a countably paracompact foliated (not necessarily smooth) manifold and G ℓ be any group of foliation-preserving homeomorphisms of ( X ℓ , Φ ℓ ) which contains the group H 0 ( X ℓ , Φ ℓ ) of all foliation-preserving homeomorphisms which take every leaf to itself. Let φ be an isomorphism between G 1 and G 2 . Then there is a foliation-preserving homeomorphism τ between X 1 and X 2 such that φ ( g ) = τ ○ g ○ τ − 1 for every g ∈ G 1 . h Theorems I and II, τ is unique.