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 \(\mathcal{H}(M,D)\) which consists of all autohomeomorphisms of \(M\) that map \(D\) onto itself equipped with the compact-open topology. The authors present a complete solution to the topological classification problem for \(\mathcal{H}(M,D)\) as follows. If \(M\) is a one-dimensional topological manifold, then they proved in an earlier paper that \(\mathcal{H}(M,D)\) is homeomorphic to \(\mathbb{Q}^\omega\), the countable power of the space of rational numbers. In all other cases they find in this paper that \(\mathcal{H}(M,D)\) is homeomorphic to the famed Erdős space \(\mathfrak E\), which consists of the vectors in Hilbert space \(\ell^2\) with rational coordinates. They obtain the second result by developing topological characterizations of Erdős space.