{Menger universal continua}
\label{ex-Menger}

The Menger universal continuum
$M^m_n$, for $1\le n<m$ \cite{Menger1926a},  can
be defined as follows \cite[pp. 121--122]{Engelking1978a}.

 Let $\mathcal F_0=\{I^m\}$.
Inductively, suppose a collection $\mathcal F_k$ of
cubes has been defined for $k\ge 0$. Divide each
cube $D\in\mathcal F_k$  into $3^{m(k+1)}$ congruent cubes
with edges of length $\frac1{3^{m(k+1)}}$. If
$\mathcal F_{k+1}(D)$ is the collection of all these
cubes that intersect $n$-dimensional faces of $D$, then let
$\mathcal F_{k+1}= \bigcup\{\,\mathcal F_{k+1}(D): D\in \mathcal F_k\,\}$.

Define $$M^m_n=\bigcap_{k=0}^\infty \left(\bigcup \mathcal F_k\right).$$

The most popular continua are  $M^2_1$, widely known as the
\gs{Sierpi\'nski universal plane curve}{Sierpinski
universal plane curve} or the \gs{Sierpi\'nski
carpet}{Sierpinski carpet} \cite{Sierpinski1916a}, and
$M^3_1$, called the
\g{Menger universal curve}
 \cite{Menger1926a}. These particular curves
are discussed separately.



 Continuum $M^{2n+1}_n$ can also be obtained as the
 inverse limit $\varprojlim(X_i, p^{i+1}_i)$ in the following way.
 Let  $\{G_1, G_2,\dots\}$ be an open base of $S^n$ consisting of
 open $n$-cells such that $\diam G_i\to 0$. Put $X_1=S^n$ and
 $X_{i+1}=(X_i\times\{0,1\})/\sim$, where relation $\sim$
 identifies points $(x,0),(x,1)$ iff  $p^i_1(x)\notin G_i$.
 The bonding maps $p^{i+1}_i$
are natural retractions \cite{Pasynkov1965a}, \cite[pp. 98--99]{Bestvina1988a}.



\rostere
\item
$M^m_n$ is the \g{universal space} in the
class of all compacta of dimension $\le n$ which embed in
$\mathbb R^m$ \cite{Stanko1971a} (\cite{Menger1926a} for
$M^m_{m-1}$).  $M^m_{m-1}$  is universal in the class of all
$(m-1)$-dimensional subsets of $\mathbb R^m$ (\cite[Problem
1.11.D]{Engelking1978a}), and  $M^{2n+1}_n$ in the class of
all $n$-dimensional separable metric spaces
\cite{Bothe1963a}.
\item\label{ex-Menger-Menger2}
 A continuum $X$ is homeomorphic to $M^{m+1}_m$ if and only
 if $X$ can be embedded in the $(m+1)$-sphere $S^{m+1}$
 in such a way that $S^{m+1}\setminus X$ has infinitely
 many components $C_1, C_2,\dots$ such that $\diam C_i\to 0$,
 $\bd C_i\cap \bd C_j=\emptyset$ for $i\ne j$, $\bd C_i$ is an
 $m$-cell for each $i$ and $\bigcup_{i=1}^\infty \bd C_i$ is dense in $X$
(see \cite{Cannon1973a}  and \cite[Theorem 6.1.2, p. 74]{Chigogidze+1995a}).
\item\label{ex-Menger-Menger3}
A continuum $X$ is homeomorphic to $M^{2n+1}_n$ if and only
if $X$ is  $n$-dimensional, \gs{$C^{n-1}$}{Ck-space} and
\gs{$LC^{n-1}$-space}{LCk-space} with the
\gs{disjoint $n$-disks property}{disjoint n-disks property}
(\gs{$DD^nP$ property}{DDnP property}) \cite[Corollary 5.2.3,
p.98]{Bestvina1988a}.

Equivalently, $X$ is homeomorphic to $M^{2n+1}_n$ if and
only if $X$ is  $n$-dimensional, $C^{n-1}$ and
$LC^{n-1}$-space such that any continuous map from a compact
space of dimension $\le n$ into   $M^{2n+1}_n$ is a uniform
limit of embeddings (\gs{Z-embeddings}{Z-embedding})
\cite[Theorem 2.3.8, p. 36]{Bestvina1988a}.
\item
\rostere
\item
If $Z\subset M^{2n+1}_n$ is a Z-set\g{Z-set}, then, for
every open neighborhood $U\subset M^{2n+1}_n$ of $Z$ and
every $\epsilon>0$, 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
