Erdîos space and homeomorphism groups of manifolds - no. 979

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.