Homeomorphism groups of manifolds and Erdos 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 $\mathcal {H}(M,D)$ 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 $\mathcal {H}(M,D)$ as follows. If $M$ is a one-dimensional topological manifold, then $\mathcal {H}(M,D)$ is homeomorphic to $\mathbb {Q}^\infty$, the countable power of the space of rational numbers. In all other cases we found 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. We obtain the second result by developing topological characterizations of Erdős space.