Homeomorphism groups of manifolds and Erdos space

Dijkstra, Jan¹ ; van Mill, Jan¹

¹ Faculteit der Exacte Wetenschappen/Afdeling Wiskunde, Vrije Universiteit, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands

Electronic research announcements of the American Mathematical Society, Tome 10 (2004), pp. 29-38.

Résumé

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.

Détail

DOI : 10.1090/S1079-6762-04-00127-1

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