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
DOI:
https://doi.org/10.1090/S1079-6762-04-00127-1
Published electronically:
April 6, 2004
MathSciNet review:
2048429
Full-text PDF Free Access
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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 Erdős 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 Erdős 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
MR Author ID:
58030
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
MR Author ID:
124825
Email:
vanmill@cs.vu.nl
Received by editor(s):
September 30, 2003
Published electronically:
April 6, 2004
Communicated by:
Krystyna Kuperberg
Article copyright:
© Copyright 2004
American Mathematical Society