{Menger universal continua}
\label{ex-Menger}
The Menger universal continuum
$M^m_n$, for $1\le n0$, there is $\delta>0$ such that if
$Z'\subset M^{2n+1}_n$ is a Z-set and $h:Z\to Z'$ is a
$\delta$-homeomorphism, then $h$ extends to an
\gs{$\epsilon$-homeomorphism}{epsilon-homeomorphism}
$h^*:M^{2n+1}_n\to M^{2n+1}_n$ such that
$h^*|M^{2n+1}_n\setminus U=\operatorname{identity}$
\cite[Theorem 3.1.1, p. 65]{Bestvina1988a}.
\item\label{ex-Menger-Menger4b}
For every $\epsilon>0$, there exists $\delta>0$
such that if $Z,Z'$ are Z-sets\g{Z-set} in $ M^{2n+1}_n$
and $h:Z\to Z'$ is a $\delta$-homeomorphism,
then there is an $\epsilon$-homeomorphism $h^*: M^{2n+1}_n\to M^{2n+1}_n$
extending $h$ \cite[Theorem 3.1.3, p. 71]{Bestvina1988a}.
\item
Every homeomorphism between Z-sets in $M^{2n+1}_n$
extends to an autohomeomorphism of
$M^{2n+1}_n$ \cite[Corollary 3.1.5, p. 72]{Bestvina1988a}.
\rosteree
It follows that $M^{2n+1}_n$ is \g{strongly locally homogeneous} and
\g{countably dense homogeneous}.
\item
If $m<2n+1$, then $M^m_n$ is not homogeneous
\cite{Lewis1987a} (for $m\ge 2n+1$ the homogeneity of $M^m_n$
follows from the property above).
\item
$M^{2n+1}_n\times X$ is not 2-homogeneous for an arbitrary
continuum $X$ \cite{Kuperberg+1995a}.
\item\label{ex-Menger-autohomeo}
The groups of autohomeomorphisms of $M^{2n+1}_n$ and
$M^{n+1}_n$ are Polish and one-dimensional
(see \cite[Corollary 6]{Oversteegen+1994a}
and Property~\ref{ex-Menger-Menger2}).
The group is totally disconnected for $M^{2n+1}_n$
(see \cite[Theorem 1.3]{Brechner1966a} for $n=1$
and its natural extension for any $n$).
\item
The Polish group of autohomeomorphisms of $M^m_n$ is
at most one-dimensional \cite{Oversteegen+1994a}.
\item
The group of autohomeomorphisms of $M^{2n+1}_n$ is
\gs{simple}{simple group} (see \cite[Theorem
3.2.4]{Chigogidze+1995a}).
\item
Every autohomeomorphism of $M^{2n+1}_n$ is
a composition of two homeomorphisms,
each of which is the
identity on some nonempty open set \cite{Sakai1994a}.
\item
For each $t\in[n,2n+1]$, there is a copy of
$M^{2n+1}_n\subset \mathbb R^{2n+1}$ with the
\g{Hausdorff dimension} $t$ \cite[Theorem
4.2.6]{Chigogidze+1995a}.
\item
Every Z-set in $M^{2n+1}_n$ is the fixed point set of
some autohomeomorphisms of $M^{2n+1}_n$ \cite{Sakai1997a}.
\item
Any $C^{n-1}$ and $LC^{n-1}$-continuum is the image of
$M^{2n+1}_n$ under a \gs{$UV^{n-1}$-map}{UVn-map}
\cite[Theorem 5.1.8, p. 95]{Bestvina1988a}.
\item
If a Polish space $X$ contains a topological copy $M$ of
$M^{2n+1}_n$ or $M^{n+1}_n$, then the space of all copies
of $M$ in $X$ with the \g{Hausdorff metric}
is not a $G_{\delta\sigma}$-set \cite{KrupskiXXXXb}.
\rosteree