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Menger universal continua

The Menger universal continuum M^m_n$, for 1\le n<m$ [Menger 1926], can be defined as follows [Engelking 1978, pp. 121-122].

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\,\}$.


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

The most popular continua are M^2_1$, widely known as the Sierpinski universal plane curve or the Sierpinski carpet [Sierpinski 1916], and M^3_1$, called the Menger universal curve [Menger 1926]. 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$-cells such that diam_i\to 0$. 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 [Pasynkov 1965], [Bestvina 1988, pp. 98-99].

  1. M^m_n$ is the universal space in the class of all compacta of dimension \le n$ which embed in \mathbb{R}^m$ [Štanko 1971] ([Menger 1926] for M^m_{m-1}$). M^m_{m-1}$ is universal in the class of all (m-1)$-dimensional subsets of \mathbb{R}^m$ ([Engelking 1978, Problem 1.11.D]), and M^{2n+1}_n$ in the class of all n$-dimensional separable metric spaces [Bothe 1963].
  2. A continuum X$ 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_i\to 0$, \mathrm{bd}\,C_i\cap \mathrm{bd}\,C_j=\emptyset$ for i\ne j$, \mathrm{bd}\,C_i$ is an m$-cell for each i$ and \bigcup_{i=1}^\infty \mathrm{bd}\,C_i$ is dense in X$ (see [Cannon 1973] and [Chigogidze et al. 1995, Theorem 6.1.2, p. 74]).
  3. A continuum X$ is homeomorphic to M^{2n+1}_n$ if and only if X$ is n$-dimensional, C^{n-1}$ and LC^{n-1}$-space with the disjoint n$-disks property (DD^nP$ property) [Bestvina 1988, Corollary 5.2.3, p.98].

    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 (Z-embeddings) [Bestvina 1988, Theorem 2.3.8, p. 36].

    1. If Z\subset M^{2n+1}_n$ is a Z-setZ-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 \epsilon$-homeomorphism h^*:M^{2n+1}_n\to M^{2n+1}_n$ such that h^*\vert M^{2n+1}_n\setminus U=\operatorname{identity}$ [Bestvina 1988, Theorem 3.1.1, p. 65].
    2. For every \epsilon>0$, there exists \delta>0$ such that if Z,Z'$ are Z-setsZ-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$ [Bestvina 1988, Theorem 3.1.3, p. 71].

    3. Every homeomorphism between Z-sets in M^{2n+1}_n$ extends to an autohomeomorphism of M^{2n+1}_n$ [Bestvina 1988, Corollary 3.1.5, p. 72].

    It follows that M^{2n+1}_n$ is strongly locally homogeneous and countably dense homogeneous.

  4. If m<2n+1$, then M^m_n$ is not homogeneous [Lewis 1987] (for m\ge 2n+1$ the homogeneity of M^m_n$ follows from the property above).

  5. M^{2n+1}_n\times X$ is not 2-homogeneous for an arbitrary continuum X$ [Kuperberg et al. 1995].

  6. The groups of autohomeomorphisms of M^{2n+1}_n$ and M^{n+1}_n$ are Polish and one-dimensional (see [Oversteegen et al. 1994, Corollary 6] and Property 2).

    The group is totally disconnected for M^{2n+1}_n$ (see [Brechner 1966, Theorem 1.3] for n=1$ and its natural extension for any n$).

  7. The Polish group of autohomeomorphisms of M^m_n$ is at most one-dimensional [Oversteegen et al. 1994].

  8. The group of autohomeomorphisms of M^{2n+1}_n$ is simple (see [Chigogidze et al. 1995, Theorem 3.2.4]).

  9. Every autohomeomorphism of M^{2n+1}_n$ is a composition of two homeomorphisms, each of which is the identity on some nonempty open set [Sakai 1994].

  10. For each t\in[n,2n+1]$, there is a copy of M^{2n+1}_n\subset \mathbb{R}^{2n+1}$ with the Hausdorff dimension t$ [Chigogidze et al. 1995, Theorem 4.2.6].

  11. Every Z-set in M^{2n+1}_n$ is the fixed point set of some autohomeomorphisms of M^{2n+1}_n$ [Sakai 1997].

  12. Any C^{n-1}$ and LC^{n-1}$-continuum is the image of M^{2n+1}_n$ under a UV^{n-1}$-map [Bestvina 1988, Theorem 5.1.8, p. 95].

  13. 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 Hausdorff metric is not a G_{\delta\sigma}$-set [Krupski XXXXb].

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Next: Sierpinski carpet Up: Cyclic examples of locally Previous: Cyclic examples of locally
Janusz J. Charatonik, Pawel Krupski and Pavel Pyrih