Homeomorphism groups of Sierpiński carpets and Erdős space
Volume 207 / 2010
Abstract
Erdős space $\mathfrak E$ is the “rational” Hilbert space, that is, the set of vectors in $\ell^2$ with all coordinates rational. Erdős proved that $\mathfrak E$ is one-dimensional and homeomorphic to its own square $\mathfrak E \times \mathfrak E$, which makes it an important example in dimension theory. Dijkstra and van Mill found topological characterizations of $\mathfrak E$. Let $M^{n+1}_n$, $n \in \mathbb N$, be the $n$-dimensional Menger continuum in $\mathbb{R}^{n+1}$, also known as the $n$-dimensional Sierpiński carpet, and let $D$ be a countable dense subset of $M^{n+1}_n$. We consider the topological group $\mathcal{H}(M^{n+1}_n, D)$ of all autohomeomorphisms of $M^{n+1}_n$ that map $D$ onto itself, equipped with the compact-open topology. We show that under some conditions on $D$ the space $\mathcal{H}(M^{n+1}_n, D)$ is homeomorphic to $\mathfrak E$ for $n \in \mathbb{N} \setminus \{3\}$.
