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Pavel S. Gevorgyan

Publications and source records attributed to Pavel S. Gevorgyan.

At least 19 recordsLinked to original sources
arXiv2026

On the Transitive Binary $G$-Spaces

Distributive subsets of the group of all invertible continuous binary operations on a topological space are considered, and it is proved that the subgroups generated by them are also distributive. A criterion for the distributivity of a binary action of a topological group $G$ on a space $X$ is obtained. The concept of transitive binary $G$-space is introduced, and a classification of transitive distributive binary $G$-spaces is given in the case of a compact group $G$.

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arXiv2026

Binary transformation groups and topological fields

The notion of a semitransitive binary action of a group $G$ on a topological space is introduced. A duality theorem is proved, establishing a bijective correspondence between semitransitive distributive binary $G$-spaces and topological fields whose multiplicative group is isomorphic to $G$. This result yields an equivalence between the category of semitransitive distributive binary $G$-spaces and the category of topological fields with multiplicative group $G$. As applications of the duality theorem, two important results are established. It is shown that a finite group can act semitransitively, distributively, and binarily only on finite sets whose cardinality is a power of a prime number. A complete characterization of those groups that can appear as multiplicative groups of topological fields is also obtained.

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arXiv2025

Equivariant Fibrations

In this paper, we study equivariant Hurewicz fibrations, obtain their internal characteristics, and prove theorems on relationship between equivariant fibrations and fibrations generated by them. Local and global properties of equivariant fibrations are examined. An equivariant analog of the Hurewicz theorem on passing from local fibrations to global fibrations is proved. A classification of equivariant fibrations with the property of uniqueness of a covering path is given.

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arXiv2025

Bi-Equivariant Extensions Of Maps

The problem of bi-equivariant extension of continuous maps of binary $G$-spaces is considered. The concept of a structural map of distributive binary $G$-spaces is introduced, and a theorem on the bi-equivariant extension of structural maps is proven. A theorem on the bi-equivariant extension of continuous maps defined on the cross sections of a distributive binary $G$-space is also proven.

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arXiv2025

On transitive and homogeneous binary $G$-spaces

In this paper, the notions of transitivity and homogeneity in binary $G$-spaces are studied. These notions coincide for distributive binary $G$-spaces. For compact $G$, it is shown that distributive transitive binary $G$-spaces are coset spaces with a suitably defined binary $G$-action. Homogeneous binary $G$-spaces are topologically homogeneous and are separated into distinct stabilization types. Examples of each type are constructed.

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arXiv2023

Shape dimension of maps

In this paper we define a numerical shape invariant of a continuous map called shape dimension of a map, which generalizes the shape dimension of a topological space. Some basic properties and applications of this invariant are given. The question of raising the shape dimension by shape finite-dimensional maps is solved. An example of a shape finite-dimensional surjective map between shape infinite-dimensional spaces is given.

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arXiv2023

Strong Movable Categories and Strong Movability of Topological Spaces

The paper is devoted to one of the important notions of the shape theory: that of strong movability, which was primarily introduced by K. Borsuk for metrizable compacts. A strong movability criterion is proved for topological spaces, which in particular reveals a new, categorical approach to the strong movability.

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arXiv2023

Movable categories

The notion of movability for metrizable compacts was introduced by K.Borsuk. In this paper we define the notion of movable category and prove that the movability of a topological space $X$ coincides with the movability of a suitable category, which is generated by the topological space $X$ (i.e., the category $W^X$, defined by S.Mardesic).

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arXiv2023

Some questions of equivariant movability

In this paper the some questions of equivariant movability connected with substitution of acting group $G$ on closed subgroup $H$ and with transitions to spaces of $H$-orbits and $H$-fixed points spaces are investigated. In the special case the characterization of equivariant-movable $G$-spaces is given.

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arXiv2023

Uniformly Movable Categories And Uniform Movability Of Topological Spaces

A categorical generalization of the notion of movability from the inverse systems and shape theory was given by the first author who defined the notion of movable category and interpreted by this the movability of topological spaces. In this paper the authors define the notion of uniformly movable category and prove that a topological space is uniformly movable in the sense of the shape theory if and only if its comma category in the homotopy category HTop over the subcategory HPol of polyhedra is a uniformly movable category. This is a weakened version of the categorical notion of uniform movability introduced by the second author.

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arXiv2023

On equivariant fibrations of $G$-CW-complexes

It is proved that if $G$ is a compact Lie group, then an equivariant Serre fibration of $G$-CW-complexes is an equivariant Hurewicz fibration in the class of compactly generated $G$-spaces. In the nonequivariant setting, this result is due to Steinberger, West and Cauty. The main theorem is proved using the following key result: a $G$-CW-complex can be embedded as an equivariant retract in a simplicial $G$-complex. It is also proved that an equivariant map $p: E \to B$ of $G$-CW-complexes is a Hurewicz $G$-fibration if and only if the $H$-fixed point map $p^H : E^H \to B^H$ is a Hurewicz fibration for any closed subgroup $H$ of $G$. This gives a solution to the problem of James and Segal in the case of $G$-CW-complexes.

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arXiv2023

Equivariant Movability of Topological Groups

The equivariant movability of topological spaces with an action of a given topological group $G$ is considered. In particular, the equivariant movability of topological groups is studied. It is proved that a second countable group $G$ is Lie if and only if it is equivariantly movable.

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arXiv2023

Groups of Invertible Binary Operations of a Topological Space

In this paper, continuous binary operations of a topological space are studied and a criterion of their invertibility is proved. The classification problem of groups of invertible continuous binary operations of locally compact and locally connected spaces is solved. A theorem on the binary distributive representation of a topological group is also proved.

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arXiv2023

Yu. M. Smirnov's General Equivariant Shape Theory

A general equivariant shape theory for arbitrary $G$-spaces in the case of a compact group $G$ is constructed by using the method of pseudometrics suggested by Yu. M. Smirnov as early as in 1985 at the fifth Tiraspol symposium on general topology and its applications.

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arXiv2023

Shape Theory

Shape theory was founded by K.~Borsuk 50 years ago. In essence, this is spectral homotopy theory; it occupies an important place in geometric topology. The article presents the basic concepts and the most important, in our opinion, results of shape theory. Unfortunately, many other interesting problems and results related to this theory could not be covered because of space limitations. The article contains an extensive bibliography for those who wants to gain a more detailed and systematic insight into the issues considered in the survey.

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arXiv2023

Bi-Equivariant Fibrations

The lifting problem for continuous bi-equivariant maps and bi-equivariant covering homotopies is considered, which leads to the notion of a bi-equivariant fibration. An intrinsic characteristic of a bi-equivariant Hurewicz fibration is obtained. Theorems concerning a relationship between bi-equivariant fibrations and fibrations generated by them are proved.

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