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Erdős Space and Homeomorphism Groups of Manifolds
Jan J. Dijkstra and Jan van Mill, Vrije Universiteit, Amsterdam, The Netherlands

Memoirs of the American Mathematical Society
2010; 62 pp; softcover
Volume: 208
ISBN-10: 0-8218-4635-3
ISBN-13: 978-0-8218-4635-3
List Price: US$61
Individual Members: US$36.60
Institutional Members: US$48.80
Order Code: MEMO/208/979
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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.

Table of Contents

  • Introduction
  • Erdős space and almost zero-dimensionality
  • Trees and \(\mathbb{R}\)-trees
  • Semi-continuous functions
  • Cohesion
  • Unknotting Lelek functions
  • Extrinsic characterizations of Erdős space
  • Intrinsic characterizations of Erdős space
  • Factoring Erdős space
  • Groups of homeomorphisms
  • Bibliography
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