Homeomorphism groups of manifolds and Erdös space

Let M be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let D be an arbitrary countable dense subset of M. Consider the topological group H(M, D) which consists of all autohomeomorphisms of M that map D onto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem for H(M, D) as follows. If M is a one-dimensional topological manifold, then H(M, D) is homeomorphic to ℚ