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Nursultan Kuanyshov

Publications and source records attributed to Nursultan Kuanyshov.

10 recordsLinked to original sources
arXiv2026

Arboricity and Simplicial Geometric Category of Wedges and Joins of Graphs

We investigate the behavior of arboricity under two fundamental graph operations, namely wedges and joins, proving an exact formula for wedges and establishing general upper and lower bounds for joins. Using the characterization of the simplicial geometric category of connected graphs in terms of arboricity, we derive a wedge formula for simplicial geometric category and obtain corresponding estimates for graph joins. Finally, we illustrate these results through explicit computations for several classes of graphs by constructing forest decompositions and the associated covers by strongly collapsible subcomplexes.

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arXiv2026

m-Contiguity Distance

In this paper, we systematically develop the $m$-contiguity distance between simplicial maps as a discrete approximation framework for homotopical complexity in the category of simplicial complexes. We construct an increasing sequence of invariants that approximate the contiguity distance from below. We prove that $m$-contiguity distance is invariant under strong homotopy equivalence and that $m$-contiguity distance coincides with the usual contiguity distance provided that the dimension of the domain simplicial complex is $m$. The fundamental properties of $m$-contiguity distance are established, including its behaviour under barycentric subdivision, under compositions, and a categorical poduct inequality. As applications of this theory, we define the $m$-simplicial Lusternik-Schnirelmann category and the $m$-discrete topological complexity, proving that each arises naturally as a special case of $m$-contiguity distance. We also showed that $SD_1(φ,ψ)=SD(φ,ψ)$ under some conditions related to aspherical spaces.

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arXiv2026

Sequential $m$-contiguity distance

In this paper, we introduce the notion of sequential $m$-contiguity distance for finitely many simplicial maps as a higher analogue of contiguity distance. This invariant generalizes both higher contiguity distance and $m$-contiguity distance, and provides a combinatorial counterpart of sequential $m$-homotopic distance. We investigate its fundamental properties, including invariance under strong homotopy type, behaviour under compositions, categorical products, and barycentric subdivision. Moreover, we define sequential $m$-discrete topological complexity of simplicial complexes. As applications, we characterise this invariant (along with $m$-simplicial LS category) in terms of sequential $m$-contiguity distance and prove that they are invariants of strong homotopy type. Furthermore, we establish inequalities relating $m$-simplicial LS category and $m$-discrete sequential topological complexity, extending classical results from topological complexity theory to the simplicial and $m$-dimensional setting.

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arXiv2026

Cohomological dimension of a Lie algebra homomorphism

We introduce the notions of the cohomological dimension and the homological dimension of Lie algebra homomorphisms, extending the corresponding invariants for group homomorphisms. We establish their basic properties, including their behavior under monomorphisms, epimorphisms, pullbacks, composition, and restriction to the image. We also obtain a chain homotopy characterization of the cohomological dimension and compute these invariants explicitly for homomorphisms between free and abelian Lie algebras.

math.RA↗
arXiv2025

On numerical invariants of retraction map

We introduce a notion of retraction between continuous maps of topological spaces and study the behavior of several numerical invariants under such retractions. These include (co)homological dimensions, the Lusternik-Schnirelmann category, the topological complexity, and the sequential topological complexity. We prove that, under the retraction map, the corresponding inequalities between invariants hold. Our results also apply to recent invariants are defined by Dranishnikov, Jauhari \cite{DJ} and Knudsen, Weinberger \cite{KW}.

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arXiv2024

On the sequential topological complexity of group homomorphisms

We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f:X\to Y,$ denoted $TC_{r}(f)$, that interacts with $TC_{r}(X)$ and $TC_{r}(Y)$ in the same way Jamie Scott's topological complexity map $TC(f)$ interacts with $TC(X)$ and $TC(Y).$ Furthermore, we apply $TC_{r}(f)$ to studying group homomorphisms $ϕ: Γ\to Λ.$ In addition, we prove that the sequential topological complexity of any nonzero homomorphism of a torsion group cannot be finite. Also, we give the characterisation of cohomological dimension of group homomorphisms.

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arXiv2023

On the LS-category of homomorphism of almost nilpotent groups

We prove the equality $\cat(ϕ)=\cd(ϕ)$ for epimorphisms $ϕ:Γ\to Λ$ between torsion-free, finitely generated almost nilpotent groups $Γ$ and $Λ$. In addition, we prove the equality $\cat(ϕ)=\cd(ϕ)$ for homomorphisms $ϕ:Γ\to Λ$ between torsion-free, finitely generated virtually nilpotent groups.

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arXiv2022

On the LS-category of homomorphisms

We prove the equality $\cat(ϕ)=\cd(ϕ)$ for homomorphisms $ϕ:Γ\to Λ$ of a torsion free finitely generated nilpotent groups $Γ$ to an arbitrary group $Λ$. We construct an epimorphism $ψ:G\to H$ between geometrically finite groups with $\cat(ψ)> \cd(ψ)$.

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