arXiv subjects

Behrooz Mashayekhy

Publications and source records attributed to Behrooz Mashayekhy.

At least 19 recordsLinked to original sources

arXiv2026

Comparison of Topologies on Homotopy Groups with Subgroup Topology Viewpoint

By introducing various topologies on the homotopy groups of a topological space, some researchers make these well known notions in algebraic topology more useful and powerful. In this paper, first we recall and review some known topologies on homotopy groups. Then by reviewing some famous subgroups of homotopy groups and using the concept of subgroup topology, we intend to compare these topologies in order to present some results on topologized homotopy groups.

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arXiv2026

Covering Maps with respect to Topologies on the Fundamental Group

In this paper, using the classical covering theory, we introduce a generalization of covering maps of a space $X$ with respect to a topology $\tau$ on the fundamental group of $X$. We show that the famous notions, covering, semicovering, generalized covering and fibration maps are of special cases of this new notion $\pi_1^{\tau}$-covering map. Moreover, among presenting some properties for this new notion, we compare $\pi_1^{\tau}$-covering maps of a space $X$ for several famous topologies on the fundamental group of $X$.

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arXiv2025

Comparison of Topologies on Fundamental Groups with Subgroup Topology Viewpoint

In order to make the fundamental group, one of the most well known invariants in algebraic topology, more useful and powerful some researchers have introduced and studied various topologies on the fundamental group from the beginning of the 21st century onwards. In this paper by reviewing these topologies, using the concept of subgroup topology, we are going to compare these topologies in order to present some results on topologized fundamental groups.

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arXiv2024

A Positive Answer to a Question of K. Borsuk on the Capacity of Polyhedra with Finite by Cyclic Fundamental Group

Karol Borsuk in 1968 asked: Is it true that every finite polyhedron dominates only finitely many different shapes? Danuta Kolodziejczyk showed that generally an answer to the Borsuk question is negative and also presented a positive answer by proving that every polyhedron with finite fundamental group dominates only finitely many different homotopy types (hence shapes). In this paper, we show that polyhedra with finite by cyclic fundamental group dominate only finitely many homotopy types. As a consequence, we give a partial positive answer to this question of Kolodziejczyk: Does every polyhedron with abelian fundamental group dominate only finitely many different homotopy types? In fact, we that every polyhedron with abelian fundamental group of rank 1 dominates only finitely many different homotopy types. Finally, we prove that every polyhedron dominates only finitely many homotopy types of simply connected CW-complexes.

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arXiv2023

On Upper Bounds for the Depth of some Classes of Polyhedra

In this paper, we present upper bounds for the depth of some classes of polyhedra, including: polyhedra with finite fundamental group, polyhedra $P$ with abelian or free $π_1(P)$ and finitely generated $H_i(tilde{P};\mathbb{Z}$, 2-dimensional polyhedra with abelian or free fundamental group, and 2-dimensional polyhedra with elementary amenable fundamental group $G$ with finite cohomological dimension $cd(G)$. Furthermore, we provide some examples to show that some of these bounds are sharp.

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arXiv2022

On Topological Homotopy Groups and Relation to Hawaiian Groups

By generalizing the whisker topology on the $n$th homotopy group of pointed space $(X, x_0)$, denoted by $π_n^{wh}(X, x_0)$, we show that $π_n^{wh}(X, x_0)$ is a topological group if $n \ge 2$. Also, we present some necessary and sufficient conditions for $π_n^{wh}(X,x_0)$ to be discrete, Hausdorff and indiscrete. Then we prove that $L_n(X,x_0)$ the natural epimorphic image of the Hawaiian group $\mathcal{H}_n(X, x_0)$ is equal to the set of all classes of convergent sequences to the identity in $π_n^{wh}(X, x_0)$. As a consequence, we show that $L_n(X, x_0) \cong L_n(Y, y_0)$ if $π_n^{wh}(X, x_0) \cong π_n^{wh}(Y, y_0)$, but the converse does not hold in general, except for some conditions. Also, we show that on some classes of spaces such as semilocally $n$-simply connected spaces and $n$-Hawaiian like spaces, the whisker topology and the topology induced by the compact-open topology of $n$-loop space coincide. Finally, we show that $n$-SLT paths can transfer $π_n^{wh}$ and hence $L_n$ isomorphically along its points.

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arXiv2021

On quasi-small loop groups

In this paper, we study some properties of homotopical closeness for paths. We define the quasi-small loop group as the subgroup of all classes of loops that are homotopically close to null-homotopic loops, denoted by $π_1^{qs} (X, x)$ for a pointed space $(X, x)$. Then we prove that, unlike the small loop group, the quasi-small loop group $π_1^{qs}(X, x)$ does not depend on the base point, and that it is a normal subgroup containing $π_1^{sg}(X, x)$, the small generated subgroup of the fundamental group. Also, we show that a space $X$ is homotopically path Hausdorff if and only if $π_1^{qs} (X, x)$ is trivial. Finally, as consequences, we give some relationships between the quasi-small loop group and the quasi-topological fundamental group.

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arXiv2021

On Nilpotent Multipliers of Pairs of Groups

In this paper, we determine the structure of the nilpotent multipliers of all pairs $(G,N)$ of finitely generated abelian groups where $N$ admits a complement in $G$. Moreover, some inequalities for the nilpotent multipliers of pairs of finite groups and their factor groups are given.

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arXiv2019

On the Capacity and Depth of Compact Surfaces

K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of capacity and depth of a compactum. In this paper, we compute the capacity and depth of compact surfaces. We show that the capacity and depth of every compact orientable surface of genus $g\geq 0$ is equal to $g+2$. Also, we prove that the capacity and depth of a compact non-orientable surface of genus $g>0$ is $[\frac{g}{2}]+2$.

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arXiv2017

On Exact Sequences of the Rigid Fibrations

In 2002, Biss investigated on a kind of fibration which is called rigid covering fibration (we rename it by rigid fibration) with properties similar to covering spaces. In this paper, we obtain a relation between arbitrary topological spaces and its rigid fibrations. Using this relation we obtain a commutative diagram of homotopy groups and quasitopological homotopy groups and deduce some results in this field.

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arXiv2017

An Upper Bound for the Depth of Some Classes of Polyhedra

K. Borsuk in the seventies introduced the notions of capacity and depth of compacta together with some relevant problems. In this paper, first, we introduce the concepts of the (strong) capacity and the (strong) depth of an object in an arbitrary category. Then in the category of groups, we compute the (strong) capacity and the (strong) depth of some well-known groups. Finally, we find an upper bound for the depth of some classes of finite polyhedra which generalizes a result of D. Kolodziejczyk in this subject.

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arXiv2017

The Capacity of Some Classes of Polyhedra

K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the concept of the capacity of a compactum. In this paper, we compute the capacity of the product of two spheres of the same or different dimensions and the capacity of lense spaces. Also, we present an upper bound for the capacity of a $\mathbb{Z}_n$-complex, i.e., a connected finite 2-dimensional CW-complex with finite cyclic fundamental group $\mathbb{Z}_n$.

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arXiv2017

On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, $π_{1}^{qtop}(X,x_{0})$ and $π_{1}^{wh}(X,x_{0})$, respectively. In particular, we give necessary or sufficient conditions for coincidence and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the reverse of some of these implications do not hold, in general.

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arXiv2017

On a Van Kampen Theorem for Hawaiian Groups

The paper is devoted to study the $n$th Hawaiian group $\mathcal{H}_n$, $n \ge 1$, of the wedge sum of two spaces $(X,x_*) = (X_1, x_1) \vee (X_2, x_2)$. Indeed, we are going to give some versions of the van Kampen theorem for Hawaiian groups of the wedge sum of spaces. First, among some results on Hawaiian groups of semilocally strongly contractible spaces, we present a structure for the $n$th Hawaiian group of the wedge sum of CW-complexes. Second, we give more informative structures for the $n$th Hawaiian group of the wedge sum $X$, when $X$ is semilocally $n$-simply connected at $x_*$. Finally, as a consequence, by generalizing the well-known Griffiths space for dimension $n\geq 1$, we give some information about the structure of Hawaiian groups of Griffiths spaces at any points.

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arXiv2017

Adjointness of Suspension and Shape Path Functors

In this paper, we introduce a subcategory $\widetilde{Sh}_*$ of Sh$_*$ and obtain some results in this subcategory. First we show that there is a natural bijection $Sh (Σ(X, x), (Y,y))\cong Sh((X,x),Sh((I, \dot{I}),(Y,y)))$, for every $(Y,y)\in \widetilde{Sh}_*$ and $(X,x)\in Sh_*$. By this fact, we prove that for any pointed topological space $(X,x)$ in $\widetilde{Sh}_*$, $\checkπ_n^{top}(X,x)\cong \checkπ_{n-k}^{top}(Sh((S^k, *),(X,x)), e_x)$, for all $1\leq k \leq n-1$.

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arXiv2017

On the Order of the Schur Multiplier of a Pair of Finite p-Groups II

Let $G$ be a finite $p$-group and $N$ be a normal subgroup of $G$, with $|N|=p^n$ and $|G/N|=p^m$. A result of Ellis (1998) shows that the order of the Schur multiplier of such a pair $(G,N)$ of finite $p$-groups is bounded by $ p^{\frac{1}{2}n(2m+n-1)}$ and hence it is equal to $ p^{\frac{1}{2}n(2m+n-1)-t}$, for some non-negative integer $t$. Recently the authors characterized the structure of $(G,N)$ when $N$ has a complement in $G$ and $t\leq 3$. This paper is devoted to classify the structure of $(G,N)$ when $N$ has a normal complement in $G$ and $t=4,5$.

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arXiv2017

On Varietal Capability of Infinite Direct Products of Groups

Recently, the authors gave some conditions under which a direct product of finitely many groups is $\mathcal{V}-$capable if and only if each of its factors is $\mathcal{V}-$capable for some varieties $\mathcal{V}$. In this paper, we extend this fact to any infinite direct product of groups. Moreover, we conclude some results for $\mathcal{V}-$capability of direct products of infinitely many groups in varieties of abelian, nilpotent and polynilpotent groups.

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arXiv2017

On h-Fibrations

In this paper, we study h-fibrations, a weak homotopical version of fibrations which have weak covering homotopy property. We present some homotopical analogue of the notions related to fibrations and characterize h-fibrations using them. Then we construct some new categories by h-fibrations and deduce some results in these categories such as the existence of products and coproducts.

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