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Vesko Valov

Publications and source records attributed to Vesko Valov.

At least 37 records · Page 2Linked to original sources
arXiv2017

Local homological properties and cyclicity of homogeneous ANR compacta

In accordance with the Bing-Borsuk conjecture \cite{bb}, 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 homological properties of \bar U and bdU are similar to the properties of the closed ball B^n in R^n and its boundary S^{n-1}. We discuss also the following questions raised by Bing-Borsuk, where X is a homogeneous ANR-compactum with dim X=n: Is it true that $X$ is cyclic in dimension n? Is it true that no non-empty closed subset of X, acyclic in dimension n-$, separates X? It is shown that both questions have positive answers simultaneously, and a positive solution to each one of them implies a solution to another question of Bing-Borsuk (whether every finite-dimensional homogenous metric AR-compactum is a point).

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arXiv2017

On homologically locally connected spaces

We provide some properties and characterizations of homologically $UV^n$-maps and $lc^n_G$-spaces. We show that there is a parallel between recently introduced by Cauty algebraic $ANR$'s and homologically $lc^n_G$-metric spaces, and this parallel is similar to the parallel between ordinary $ANR$'s and $LC^n$-metric spaces. We also show that there is a similarity between the properties of $LC^n$-spaces and $lc^n_G$-spaces. Some open questions are raised.

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arXiv2017

Homological selections and fixed-point theorems

A homological selection theorem for C-spaces, as well as, a finite-dimensional homological selection theorem is established. We apply the finite-dimensional homological selection theorem to obtain fixed-point theorems for usco homologically UV^n set-valued maps.

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arXiv2017

Homological dimension and homogeneous ANR spaces

The homological dimension $d_G$ of metric compacta was introduced by Alexandroff. In this paper we provide some general properties of $d_G$, mainly with an eye towards describing the dimensional full-valuedness of compact metric spaces. As a corollary of the established properties of $d_G$, we prove that any two-dimensional $lc^2$ metric compactum is dimensionally full-valued. This improves the well known result of Kodama that every two-dimensional $ANR$ is dimensionally full-valued. Applications for homogeneous metric $ANR$-compacta are also given.

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arXiv2017

Homological dimension and dimensional full-valuedness

There are different definitions of homological dimension of metric compacta involving either Čech homology or exact (Steenrod) homology. In this paper we investigate the relation between these homological dimensions with respect to different groups. It is shown that all homological dimensions of a metric compactum X with respect to any field coincide provided X is homologically locally connected with respect to the singular homology up to dimension n=dim X. We also prove that any two-dimensional lc^2 metric compactum X satisfies the equality dim(X times Y)=dim X+dim Y for any metric compactum Y. This improves the well known result of Kodama that every two-dimensional ANR is dimensionally full-valued. Actually, the condition X to be lc^2 can be weaken to the existence at every point x a neighborhood V of x such that the inclusion homomorphism H_k(V;S^1)\to H_k(X;S^1)$ is trivial for all k=1,2.

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arXiv2016

Parametric set-wise injective maps

We introduce the notion of set-wise injective maps and provide results about fiber embeddings. Our results improve some previous results in this area.

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arXiv2015

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 compactum is dimensionally full-valued.

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arXiv2015

Game theoretic approach to skeletally Dugundji and Dugundji spaces

Characterizations of skeletally Dugundji spaces and Dugundji spaces are given in terms of club collections, consisting of countable families of co-zero sets. For example, a Tychonoff space $X$ is skeletally Dugundji if and only if there exists an additive $c$-club on $X$. Dugundji spaces are characterized by the existence of additive $d$-clubs.

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arXiv2014

Locally $n$-connected compacta and $UV^n$-maps

We provide a machinery for transferring some properties of metrizable $ANR$-spaces to metrizable $LC^n$-spaces. As a result, we show that for complete metrizable spaces the properties $ALC^n$, $LC^n$ and $WLC^n$ coincide to each other. We also provide the following spectral characterizations of $ALC^n$ and cell-like compacta: A compactum $X$ is $ALC^n$ if and only if $X$ is the limit space of a $σ$-complete inverse system $S=\{X_α, p^β_α, α<β<τ\}$ consisting of compact metrizable $LC^n$-spaces $X_α$ such that all bonding projections $p^β_α$, as a well all limit projections $p_α$, are $UV^n$-maps. A compactum $X$ is a cell-like (resp., $UV^n$) space if and only if $X$ is the limit space of a $σ$-complete inverse system consisting of cell-like (resp., $UV^n$) metric compacta.

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arXiv2013

A selection theorem for set-valued maps into normally supercompact spaces

The following selection theorem is established:\\ Let $X$ be a compactum possessing a binary normal subbase $\mathcal S$ for its closed subsets. Then every set-valued $\mathcal S$-continuous map $Φ\colon Z\to X$ with closed $\mathcal S$-convex values, where $Z$ is an arbitrary space, has a continuous single-valued selection. More generally, if $A\subset Z$ is closed and any map from $A$ to $X$ is continuously extendable to a map from $Z$ to $X$, then every selection for $Φ|A$ can be extended to a selection for $Φ$. This theorem implies that if $X$ is a $κ$-metrizable (resp., $κ$-metrizable and connected) compactum with a normal binary closed subbase $\mathcal S$, then every open $\mathcal S$-convex surjection $f\colon X\to Y$ is a zero-soft (resp., soft) map. Our results provide some generalizations and specifications of Ivanov's results (see \cite{i1}, \cite{i2}, \cite{i3}) concerning superextensions of $κ$-metrizable compacta.

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arXiv2012

Alexandroff Manifolds and Homogeneous Continua

We prove the following result announced in Todorov and Valov: Any homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum with $\check{H}^n(X;G)\neq 0$ is a $V^n$-continuum in the sense of Alexandroff (1957). We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu (2007) whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs.

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arXiv2011

On a theorem of Arvanitakis

Arvanitakis established recently a theorem which is a common generalization of Michael's convex selection theorem and Dugundji's extension theorem. In this note we provide a short proof of a more general version of Arvanitakis' result.

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arXiv2011

Functional extenders and set-valued retractions

We describe the supports of a class of real-valued maps on $C*(X)$ introduced by Radul. Using this description, a characterization of compact-valued retracts of a given space in terms of functional extenders is obtained. For example, if $X\subset Y$, then there exists a continuous compact-valued retraction from $Y$ onto $X$ if and only if there exists a normed weakly additive extender $u\colon C*(X)\to C*(Y)$ with compact supports preserving $\min$ (resp., $\max$) and weakly preserving $\max$ (resp., $\min$). Similar characterizations are obtained for upper (resp., lower) semi-continuous compact-valued retractions. These results provide characterizations of (not necessarily compact) absolute extensors for zero-dimensional spaces, as well as absolute extensors for one-dimensional spaces, involving non-linear functional extenders.

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arXiv2011

Another approach to parametric Bing and Krasinkiewicz maps

Using a factorization theorem due to Pasynkov we provide a short proof of the existence and density of parametric Bing and Krasinkiewicz maps. In particular, the following corollary is established: Let $f\colon X\to Y$ be a surjective map between paracompact spaces such that all fibers $f^{-1}(y)$, $y\in Y$, are compact and there exists a map $g\colon X\to\mathbb I^{\aleph_0}$ embedding each $f^{-1}(y)$ into $\mathbb I^{\aleph_0}$. Then for every $n\geq 1$ the space $C^*(X,\mathbb R^n)$ of all bounded continuous functions with the uniform convergence topology contains a dense set of maps $g$ such that any restriction $g|f^{-1}(y)$, $y\in Y$, is a Bing and Krasinkiewicz map.

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arXiv2011

Maps with dimensionally restricted fibers

We prove that if $f\colon X\to Y$ is a closed surjective map between metric spaces such that every fiber $f^{-1}(y)$ belongs to a class of space $\mathrm S$, then there exists an $F_σ$-set $A\subset X$ such that $A\in\mathrm S$ and $\dim f^{-1}(y)\backslash A=0$ for all $y\in Y$. Here, $\mathrm S$ can be one of the following classes: (i) $\{M:\mathrm{e-dim}M\leq K\}$ for some $CW$-complex $K$; (ii) $C$-spaces; (iii) weakly infinite-dimensional spaces. We also establish that if $\mathrm S=\{M:\dim M\leq n\}$, then $\dim f\triangle g\leq 0$ for almost all $g\in C(X,\mathbb I^{n+1})$.

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arXiv2010

External characterization of I-favorable spaces

We provide both a spectral and an internal characterizations of arbitrary I-favorable spaces with respect to co-zero sets. As a corollary we establish that any product of compact I-favorable spaces with respect to co-zero sets is also I-favorable with respect to co-zero sets. We also prove that every C*-embedded I-favorable with respect to co-zero sets subspace of an extremally disconnected space is extremally disconnected.

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