On the reconstruction of topological spaces from their groups of homeomorphisms

Abstract:

For various classes $K$ of topological spaces we prove that if ${X_1},{X_2} \in K$ and ${X_1},{X_2}$ have isomorphic homeomorphism groups, then ${X_1}$ and ${X_2}$ are homeomorphic. Let $G$ denote a subgroup of the group of homeomorphisms $H(X)$ of a topological space $X$. A class $K$ of $\langle X,G\rangle$’s is faithful if for every $\langle {X_1},{G_1}\rangle ,\langle {X_2},{G_2}\rangle \in K$, if $\varphi :{G_1} \to {G_2}$ is a group isomorphism, then there is a homeomorphism $h$ between ${X_1}$ and ${X_2}$ such that for every $g \in {G_1}\;\varphi (g) = hg{h^{ - 1}}$. Theorem 1: The following class is faithful: $\{ \langle X,H(X)\rangle |(X$ is a locally finite-dimensional polyhedron in the metric or coherent topology or $X$ is a Euclidean manifold with boundary) and for every $x \in X\;x$ is an accumulation point of $\{ g(x)|g \in H(X)\} \} \cup \{ \langle X,G\rangle |X$ is a differentiable or a $PL$-manifold and $G$ contains the group of differentiable or piecewise linear homeomorphisms$\}$ $\cup \{ \langle X,H(X)\rangle |X$ is a manifold over a normed vector space over an ordered field$\}$. This answers a question of Whittaker $[{\text {W}}]$, who asked about the faithfulness of the class of Banach manifolds. Theorem 2: The following class is faithful: $\{ \langle X,G\rangle |X$ is a locally compact Hausdorff space and for every open $T \subseteq X$ and $x \in T\;\{ g(x)|g \in H(X)$ and $g \upharpoonright (X - T) = \operatorname {Id}\}$ is somewhere dense$\}$. Note that this class includes Euclidean manifolds as well as products of compact connected Euclidean manifolds. Theorem 3: The following class is faithful: $\{ \langle X,H(X)\rangle |$ (1) $X$ is a $0$-dimensional Hausdorff space; (2) for every $x \in X$ there is a regular open set whose boundary is $\{ x\}$; (3) for every $x \in X$ there are ${g_{1,}}{g_2} \in G$ such that $x \ne {g_1}(x) \ne {g_2}(x) \ne x$, and (4) for every nonempty open $V \subseteq X$ there is $g \in H(X) - \{ \operatorname {Id}\}$ such that $g \upharpoonright (X - V) = \operatorname {Id}\}$. Note that (2) is satisfied by $0$-dimensional first countable spaces, by order topologies of linear orderings, and by normed vector spaces over fields different from ${\mathbf {R}}$. Theorem 4: We prove (Theorem 2.23.1) that for an appropriate class ${K^T}$ of trees $\{ \langle \operatorname {Aut}(T),T; \leq , \circ ,\operatorname {Op}\rangle |T \in {K^T}\}$ is first-order interpretable in $\{ \operatorname {Aut}(T)|T \in {K^T}\}$.