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 . We say that G is a factorizable group of homeomorphisms, if for every open cover U of X, generates G.Theorem I – Let 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 II – For ℓ=1,2 let (Xℓ,Φℓ) be a countably paracompact foliated (not necessarily smooth) manifold and Gℓ be any group of foliation-preserving homeomorphisms of (Xℓ,Φℓ) 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.