Homeomorphism Groups of Manifolds and Erdős 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∞, the countable power of the space of rational numbers. In all other cases we found that H(M,D) is homeomorphic to the famed Erdős space E, which consists of the vectors in Hilbert space `2 with rational coordinates. We obtain the second result by developing topological characterizations of Erdős space.