Universal spaces for asymptotic dimension via factorization

2005

Gromov \cite{Gr$_1$} and Dranishnikov \cite{Dr$_1$} introduced asymptotic and coarse dimensions of proper metric spaces via quite different ways. We define coarse and asymptotic dimension of all metric spaces in a unified manner and we investigate relationships between them generalizing results of Dranishnikov \cite{Dr$_1$} and Dranishnikov-Keesling-Uspienskij \cite{DKU}.

2013

Asymptotic property C for metric spaces was introduced by Dranishnikos as generalization of finite asymptotic dimension - asdim. It turns out that this property can be viewed as transfinite extension of asymptotic dimension. The original definition was given by Radul. We introduce three equivalent definitions, show that asymptotic property C is closed under products (open problem stated "Open problems in topology II") and prove some other facts, i.e. by defining dimension of a family of metric spaces. Some examples of spaces enjoying countable trasfinite asymptotic dimension are given. We also formulate open problems and state "omega conjecture", which inspired most part of this paper.

Topology and Its Applications, 2013

Given a nowheredense closed subset $X$ of a metrizable compact space $\tx$, we characterize the dimension of $X$ in terms of the multiplicity of the canonicals covers of the complementary of $X$, specially in some particular cases, like when $\tx$ is the Hilbert cube or the finite dimensional cube and $X$, a Z-set of $\tx$. In this process, we solve some questions in the literature.

Proceedings of the American Mathematical Society, 1999

Oversteegen and Tymchatyn proved that homeomorphism groups of positive dimensional Menger compacta are 1 1 -dimensional by proving that almost 0 0 -dimensional spaces are at most 1 1 -dimensional. These homeomorphism groups are almost 0 0 -dimensional and at least 1 1 -dimensional by classical results of Brechner and Bestvina. In this note we prove that almost n n -dimensional spaces for n ≥ 1 n \geq 1 are n n -dimensional. As a corollary we answer in the affirmative an old question of R. Duda by proving that every hereditarily locally connected, non-degenerate, separable, metric space is 1 1 -dimensional.

Topology and its Applications, 2007

We consider the notion of dimension in four categories: the category of (unbounded) separable metric spaces and (metrically proper) Lipschitz maps, and the category of (unbounded) separable metric spaces and (metrically proper) uniform maps. A unified treatment is given to the large scale dimension and the small scale dimension. We show that in all categories a space has dimension zero if and only if it is equivalent to an ultrametric space. Also, 0-dimensional spaces are characterized by means of retractions to subspaces. There is a universal zero-dimensional space in all categories. In the Lipschitz Category spaces of dimension zero are characterized by means of extensions of maps to the unit 0-sphere. Any countable group of asymptotic dimension zero is coarsely equivalent to a direct sum of cyclic groups. We construct uncountably many examples of coarsely inequivalent ultrametric spaces.

Topology and its Applications, 2018

It is well-known that a paracompact space X is of covering dimension at most n if and only if any map f : X → K from X to a simplicial complex K can be pushed into its n-skeleton K (n). We use the same idea to characterize asymptotic dimension in the coarse category of arbitrary coarse spaces. Continuity of the map f is replaced by variation of f on elements of a uniformly bounded cover. The same way one can generalize Property A of G.Yu to arbitrary coarse spaces.

Topology and its Applications, 2001

For any countable CW -complex K and a cardinal number τ ≥ ω we construct a completely metrizable space X(K, τ ) of weight τ with the following properties: e-dim X(K, τ ) ≤ K, X(K, τ ) is an absolute extensor for all normal spaces Y with e-dim Y ≤ K, and for any completely metrizable space Z of weight ≤ τ and e-dim Z ≤ K the set of closed embeddings Z → X(K, τ ) is dense in the space C(Z, X(K, τ )) of all continuous maps from Z into X(K, τ ) endowed with the limitation topology. This result is applied to prove the existence of universal spaces for all metrizable spaces of given weight and with a given cohomological dimension.

Mediterranean Journal of Mathematics, 2018

We introduce the notion of (hybrid) large scale normal space and prove coarse geometric analogues of Urysohn's Lemma and the Tietze Extension Theorem for these spaces, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which we obtain both the classical topological results and the (hybrid) large scale results as corollaries. We prove that all metric spaces are hybrid large scale normal, and characterize those locally compact abelian groups which (as hybrid large scale spaces) are hybrid large scale normal. Finally, we look at some properties of the Higson compactifications and coronas of hybrid large scale normal spaces.

2000

The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results: Generalized Eilenberg-Borsuk Theorem. Let L be a countable CW com- plex. If X is a separable metrizable space and KL is an absolute extensor of X for some CW complex K, then for any map f : A! K, A closed in X, there is an extension f0 : U ! K of f over an open set U such that L2 AE(X U). Theorem. Let K;L be countable CW complexes. If X is a separable metriz- able space and KL is an absolute extensor of X ,t hen there is as ubsetY of X such that K2 AE(Y ) and L2 AE(X Y ). Theorem. Suppose Gi;::: ;Gn are countable, non-trivial, abelian groups and k> 0. For any separable metrizable space X of nite dimension dim X> 0, there is a closed subset Y of X with dimGiY =m ax(dimGiX k;1) for i =1 ;::: ;n. Theorem. Suppose W is a separable metrizable space of nite dimension and Y is a compactum of nite dimension. Then...

Pacific Journal of Mathematics, 2002

A nonnegative number d∞, called asymptotic dimension, is associated with any metric space. Such number detects the asymptotic properties of the space (being zero on bounded metric spaces), fulfills the properties of a dimension, and is invariant under rough isometries. It is then shown that for a class of open manifolds with bounded geometry the asymptotic dimension coincides with the 0-th Novikov-Shubin number α0 defined in a previous paper [D. Guido, T. Isola, J. Funct. Analysis, 176 (2000)]. Thus the dimensional interpretation of α0 given in the mentioned paper in the framework of noncommutative geometry is established on metrics grounds. Since the asymptotic dimension of a covering manifold coincides with the polynomial growth of its covering group, the stated equality generalises to open manifolds a result by Varopoulos.