Memoirs of the American Mathematical Society 2010; 62 pp; softcover Volume: 208 ISBN10: 0821846353 ISBN13: 9780821846353 List Price: US$61 Individual Members: US$36.60 Institutional Members: US$48.80 Order Code: MEMO/208/979
 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 compactopen topology. The authors present a complete solution to the topological classification problem for \(\mathcal{H}(M,D)\) as follows. If \(M\) is a onedimensional 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. Table of Contents  Introduction
 Erdős space and almost zerodimensionality
 Trees and \(\mathbb{R}\)trees
 Semicontinuous functions
 Cohesion
 Unknotting Lelek functions
 Extrinsic characterizations of Erdős space
 Intrinsic characterizations of Erdős space
 Factoring Erdős space
 Groups of homeomorphisms
 Bibliography
