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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. We 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 $\mathcal{H}(M,D)$ is homeomorphic to $\mathbb{Q}^\infty$, the countable power of the space of rational numbers. In all other cases we found that $\mathcal{H}(M,D)$ is homeomorphic to the famed Erd\H os space $\mathfrak E$, which consists of the vectors in Hilbert space $\ell^2$ with rational coordinates. We obtain the second result by developing topological characterizations of Erd\H os space.

Copyright 2004 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 10 (2004), pp. 29-38
• Publisher Identifier: S 1079-6762(04)00127-1
• 2000 Mathematics Subject Classification. Primary 57S05
• Received by editors September 30, 2003
• Posted on April 6, 2004
• Communicated by Krystyna Kuperberg
• Comments (When Available)

Jan J. Dijkstra

Faculteit der Exacte Wetenschappen / Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

E-mail address: dijkstra@cs.vu.nl

Jan van Mill

Faculteit der Exacte Wetenschappen  /  Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

E-mail address: vanmill@cs.vu.nl