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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the reconstruction of topological spaces from their groups of homeomorphisms


Author: Matatyahu Rubin
Journal: Trans. Amer. Math. Soc. 312 (1989), 487-538
MSC: Primary 54H99; Secondary 20F38, 58B99
MathSciNet review: 988881
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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 \vert(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)\vert g \in H(X)\} \} \cup \{ \langle X,G\rangle \vert 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 \vert 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 \vert X$ is a locally compact Hausdorff space and for every open $ T \subseteq X$ and $ x \in T\;\{ g(x)\vert 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 \vert$ (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 \vert T \in {K^T}\} $ is first-order interpretable in $ \{ \operatorname{Aut}(T)\vert T \in {K^T}\} $.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1989-0988881-4
PII: S 0002-9947(1989)0988881-4
Article copyright: © Copyright 1989 American Mathematical Society