Isomorphisms between groups of homeomorphisms (Q1851074)

Three axioms are given for groups of homeomorphisms of manifolds which guarantee that given two manifolds \(M_1\) and \(M_2\) and respective homeomorphism groups \(G(M_1)\) and \(G(M_2)\), with any isomorphism \(\Phi:G(M_1)\to G(M_2)\) there is associated a unique homeomorphism \(\varphi:M_1\to M_2\) such that \(\Phi(h)=\varphi h\varphi^{-1}\) for each \(h\in G(M_1)\). While the term ``manifold'' is not specified it appears that the argument works for any locally Euclidean, Hausdorff space. Smooth versions also apply.