Homeomorphism groups of manifolds and Erdos space

Abstract

Let M M be either a topological manifold, a Hilbert cube manifold, or a Menger manifold and let D D be an arbitrary countable dense subset of M M . Consider the topological group H ( M , D ) \mathcal {H}(M,D) which consists of all autohomeomorphisms of M M that map D D onto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem for H ( M , D ) \mathcal {H}(M,D) as follows. If M M is a one-dimensional topological manifold, then H ( M , D ) \mathcal {H}(M,D) is homeomorphic to Q ∞ \mathbb {Q}^\infty , the countable power of the space of rational numbers. In all other cases we found that H ( M , D ) \mathcal {H}(M,D) is homeomorphic to the famed Erdős space E \mathfrak E , which consists of the vectors in Hilbert space ℓ 2 \ell ^2 with rational coordinates. We obtain the second result by developing topological characterizations of Erdős space.