Reconstruction of manifolds and subsets of normed spaces from subgroups of their homeomorphism groups
Volume 435 / 2005
Abstract
\def\ods{\vskip6pt}\def\cl{\mathop{\rm cl}\nolimits}% \def\inverse{^{-1}}% \def\rfs#1{{\rm#1}}% \def\norm#1{\|#1\|}% \font\sc=plcsc10 This work concerns topological spaces of the following types: open subsets of normed vector spaces, manifolds over normed vector spaces, the closures of open subsets of normed vector spaces and some other types of topological spaces related to the above. We show that such spaces are determined by various subgroups of their auto-homeomorphism groups. Theorems 1–3 below are typical examples of the results obtained in this work. \ods \noindent{\sc Theorem 1.} {\it For a metric space $X$ let $\rfs{UC}(X)$ denote the group of all auto-homeomorphisms $h$ of $X$ such that $h$ and $h\inverse$ are uniformly continuous. Let $X$ be an open subset of a Banach space with the following property: for every $\varepsilon > 0$ there is $\delta > 0$ such that for every $u,v \in X$: if $\norm{u - v} < \delta$, then there is an arc $L \subseteq X$ connecting $u$ and $v$ such that $\rfs{diam}(L) < \varepsilon$. Suppose that the same holds for $Y$. Let $\varphi$ be a group isomorphism between $\rfs{UC}(X)$ and $\rfs{UC}(Y)$. Then there is a homeomorphism $\tau$ between $X$ and $Y$ such that $\tau$ and $\tau\inverse$ are uniformly continuous and for every $g \in \rfs{UC}(X)$, $\varphi(g) = \tau \circ g \circ \tau\inverse$. } \ods See Corollaries 5.6 and 2.26. \ods \noindent{\sc Theorem 2.} {\it Let $H(X)$ denote the group of auto-homeomorphisms of a topological space $X$. Let $X$ be a bounded open subset of a Banach space $E$, and denote by $\cl (X)$ the closure of $X$ in~$E$. Suppose that $X$ has the following properties: $(1)$ There is $d$ such that for every $u,v \in X$ there is a rectifiable arc $L \subseteq X$ connecting $u$ and $v$ such that $\rfs{length}(L) < d$; $(2)$ for every point $w$ in the boundary of $X$ and $\varepsilon > 0$, there is $\delta > 0$ such that for every $u,v \in X$: if $\norm{u - w},\norm{v - w} < \delta$, then there is an arc $L \subseteq X$ connecting $u$ and $v$ such that $\rfs{diam}(L) < \varepsilon$. Suppose that the same holds for $Y$. Let $\varphi$ be a group isomorphism between $H(\cl (X))$ and $H(\cl (Y))$. Then there is a homeomorphism $\tau$ between $\cl (X)$ and $\cl (Y)$ such that for every $g \in H(\cl (X))$, $\varphi(g) = \tau \circ g \circ \tau\inverse$.
