We rephrase Gromov's definition of Markov compacta, introduce a subclass of Markov compacta defined by one building block and study cohomological dimensions of these compacta. We show that for a Markov compactum $X$, $\dim_{\Z_{(p)}}X=\dim_{\Q}X$ for all but finitely many primes $p$ where $\Z_{(p)}$ is the localization of $\Z$ at $p$. We construct Markov compacta of arbitrarily large dimension having $\dim_{\Q}X=1$ as well as Markov compacta of arbitrary large rational dimension with $\dim_{\Z_p}X=1$ for a given $p$.

2006-11-01math.GNmath.GT

Local cohomological properties of homogeneous ANR compacta

In accordance with the Bing-Borsuk conjecture, we show that if X is an n-dimensional homogeneous metric ANR compactum and x\in X, then there is a local basis at x consisting of connected open sets U such that the cohomological properties of \overline U and bdU are similar to the properties of the closed ball \mathbb B^n\subset\mathbb R^n and its boundary \mathbb S^{n-1}. We also prove that a metric ANR compactum X of dimension n is dimensionally full-valued if and only if the group H_n(X,X\setminus x) is not trivial for some x\in X. This implies that every 3-dimensional homogeneous metric ANR …

2015-08-10math.GNmath.GT

Resolving rational cohomological dimension via a Cantor group action

By a Cantor group we mean a topological group homeomorphic to the Cantor set. We show that a compact metric space of rational cohomological dimension $n$ can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension $n$. Moreover, the action can be assumed to be free if $n=1$.

2013-09-28math.GNmath.GT

Cell-like resolutions preserving cohomological dimensions

We prove that for every compactum X with dim_Z X <= n >= 2 there is a cell-like resolution r: Z --> X from a compactum Z onto X such that dim Z <= n and for every integer k and every abelian group G such that dim_G X <= k >= 2 we have dim_G Z <=k. The latter property implies that for every simply connected CW-complex K such that e-dim X <= K we also have e-dim Z <= K.

2004-01-11math.ATmath.GN

Extension theory of infinite symmetric products

We present an approach to cohomological dimension theory based on infinite symmetric products and on the general theory of dimension called the extension dimension. The notion of the extension dimension $\ExD(X)$ was introduced by A.N.Dranishnikov \cite {D$_5$} in the context of compact spaces and CW complexes. This paper investigates extension types of infinite symmetric products $SP(L)$. One of the main ideas of the paper is to treat $\ExD(X)\leq SP(L)$ as the fundamental concept of cohomological dimension theory instead of $\dim_G(X)\leq n$. In a subsequent paper \cite{Dy$_6$} we show how p…

2004-04-19math.ATmath.GNmath.GT

On homology of complements of compact sets in Hilbert Cube

We introduce the notion of spaces with weak relative cohomology and show the acyclicity of the complement $Q\setminus X$ in the Hilbert cube $Q$ of a compactum $X$ with weak relative cohomology. As a corollary we obtain the acyclicity of the complement results when (a) $X$ is weakly infinite dimensional; (b) $X$ has finite cohomological dimension.

2016-03-22math.ATmath.GN

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Algebra of dimension theory

The dimension algebra of graded groups is introduced. With the help of known geometric results of extension theory that algebra induces all known results of the cohomological dimension theory. Elements of the algebra are equivalence classes $\dim(A)$ of graded groups $A$. There are two geometric interpretations of those equivalence classes: \linebreak 1. For pointed CW complexes $K$ and $L$, $\dim(H_\ast(K))=\dim(H_\ast(L))$ if and only if the infinite symmetric products $SP(K)$ and $SP(L)$ are of the same extension type (i.e., $SP(K)\in AE(X)$ iff $SP(L)\in AE(X)$ for all compact $X$). \lineb…

2004-04-19math.ATmath.GNmath.GT

Universal metric spaces and extension dimension

For any countable $CW$-complex $K$ and a cardinal number $τ\geqω$ we construct a completely metrizable space $X(K,τ)$ of weight $τ$ with the following properties: $\e X(K,τ)\leq K$, $X(K,τ)$ is an absolute extensor for all normal spaces $Y$ with $\e Y\leq K$, and for any completely metrizable space $Z$ of weight $\leqτ$ and $\e Z\leq K$ the set of closed embeddings $Z\to X(K,τ)$ 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 giv…

1999-08-16math.GN