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Erdős Space and Homeomorphism Groups of Manifolds
Jan J. Dijkstra and Jan van Mill, Vrije Universiteit, Amsterdam, The Netherlands
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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$58 Individual Members: US$34.80
Institutional Members: US\$46.40
Order Code: MEMO/208/979

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.

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