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Vladimer Baladze

Publications and source records attributed to Vladimer Baladze.

5 recordsLinked to original sources
arXiv2021

On Axiomatic Characterization of Alexander-Spanier Normal Homology Theory of General Topological Spaces

The Alexandroff-Čech normal cohomology theory [Mor$_1$], [Bar], [Ba$_1$],[Ba$_2$] is the unique continuous extension \cite{Wat} of the additive cohomology theory [Mil], [Ber-Mdz$_1$] from the category of polyhedral pairs $\mathcal{K}^2_{Pol}$ to the category of closed normally embedded, the so called, $P$-pairs of general topological spaces $\mathcal{K}^2_{Top}$. In this paper we define the Alexander-Spanier normal cohomology theory based on all normal coverings and show that it is isomorphic to the Alexandroff-Čech normal cohomology. Using this fact and methods developed in [Ber-Mdz$_3$] we construct an exact, the so called, Alexander-Spanier normal homology theory on the category $\mathcal{K}^2_{Top},$ which is isomorphic to the Steenrod homology theory on the subcategory of compact pairs $\mathcal{K}^2_{C}.$ Moreover, we give an axiomatic characterization of the constructed homology theory.

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arXiv2017

Strong Homology Theory of Continuous Maps

The current work is motivated by the papers $[B_3]$, $[B_6]$, $[Be]$, $[Be-Tu]$. In particular, using Theorem 3.7 of $[B_3]$ and methods developed in this paper, the spectral and strong homology groups of continuous maps were defined and studied $[B_6]$, $[Be]$, $[Be-Tu]$. In this paper we will show that strong homology groups of continuous maps are a homology type functor, which is a strong shape invariant and has the semi-continuous property. We will formulate the new axioms and the conjunction on the uniqueness of the constructed functor.

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arXiv2017

Fiber Strong Shape Theory for Topological Spaces

In the paper we construct and develop a fiber strong shape theory for arbitrary spaces over fixed metrizable space $\Bo$. Our approach is based on the method of Mardešić-Lisica and instead of resolutions, introduced by Mardešić, their fiber preserving analogues are used. The fiber strong shape theory yields the classification of spaces over $\Bo$ which is coarser than the classification of spaces over $\Bo$ induced by fiber homotopy theory, but is finer than the classification of spaces over $\Bo$ given by usual fiber shape theory.

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arXiv2017

Čech border homology and cohomology groups and some applications

In the paper the Čech border homology and cohomology groups of closed pairs of normal spaces are constructed and investigated. These groups give intrinsic characterizations of Čech homology and cohomology groups based on finite open coverings, homological and cohomological coefficients of cyclicity, small and large cohomological dimensions of remainders of Stone-Čech compactifications of metrizable spaces.

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