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{up harpoon right}(X\U)=Id}. We say that G is a factorizable group of homeomorphisms, if for every open cover U of X, ∪_U∈UGU generates G.Theorem ILet G, 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.Theorem A strengthens known theorems in which the existence of τ is concluded from the assumption of factorizability and some additional assumptions.Theorem IIFor ℓ=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 H0(Xℓ,Φℓ) of all foliation-preserving homeomorphisms which take every leaf to itself. Let φ be an isomorphism between G₁ and G₂. Then there is a foliation-preserving homeomorphism τ between X₁ and X₂ such that φ(g)=τ○g○τ^-1 for every g∈G₁.In both Theorems I and II, τ is unique.