Homeomorphism groups of manifolds and Erdos space

Abstract

LetMMbe either a topological manifold, a Hilbert cube manifold, or a Menger manifold and letDDbe an arbitrary countable dense subset ofMM. Consider the topological groupH(M,D)\mathcal {H}(M,D)which consists of all autohomeomorphisms ofMMthat mapDDonto itself equipped with the compact-open topology. We present a complete solution to the topological classification problem forH(M,D)\mathcal {H}(M,D)as follows. IfMMis a one-dimensional topological manifold, thenH(M,D)\mathcal {H}(M,D)is homeomorphic toQ∞\mathbb {Q}^\infty, the countable power of the space of rational numbers. In all other cases we found thatH(M,D)\mathcal {H}(M,D)is homeomorphic to the famed Erdős spaceE\mathfrak E, which consists of the vectors in Hilbert spaceℓ2\ell ^2with rational coordinates. We obtain the second result by developing topological characterizations of Erdős space.