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Homeomorphism groups of manifolds and Erdos space
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## Homeomorphism groups of manifolds and Erdos space

### Jan J. Dijkstra and Jan van Mill

**Abstract.**
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`

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