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ISSN 1079-6762

 

Homeomorphism groups of manifolds and Erdos space


Authors: Jan J. Dijkstra and Jan van Mill
Journal: Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 29-38
MSC (2000): Primary 57S05
Published electronically: April 6, 2004
MathSciNet review: 2048429
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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 Erdos 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 Erdos space.


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Additional Information

Jan J. Dijkstra
Affiliation: Faculteit der Exacte Wetenschappen/Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
Email: dijkstra@cs.vu.nl

Jan van Mill
Affiliation: Faculteit der Exacte Wetenschappen/Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
Email: vanmill@cs.vu.nl

DOI: http://dx.doi.org/10.1090/S1079-6762-04-00127-1
PII: S 1079-6762(04)00127-1
Received by editor(s): September 30, 2003
Published electronically: April 6, 2004
Communicated by: Krystyna Kuperberg
Article copyright: © Copyright 2004 American Mathematical Society