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Behrooz Mashayekhy

Publications and source records attributed to Behrooz Mashayekhy.

At least 55 records · Page 3Linked to original sources
arXiv2013

Topological fundamental groups and small generated coverings

This paper is devoted to study some topological properties of the SG subgroup, $π_1^{sg}(X,x)$, of the quasitopological fundamental group of a based space $(X,x)$, $\pt$, its topological properties as a subgroup of the topological fundamental group $π_1^τ(X,x)$ and its influence on the existence of universal covering of $X$. First, we introduce small generated spaces which have indiscrete topological fundamental groups and also small generated coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of the small generated coverings. Finally, by introducing the notion of semi-locally small generatedness we show that the quasitopological fundamental groups of semi-locally small generated spaces are topological groups.

math.AT↗
arXiv2012

On the Spanier Groups and Covering and Semicovering Spaces

For a connected, locally path connected space $X$, let $H$ be a subgroup of the fundamental group of $X$, $π_1(X,x)$. We show that there exists an open cover $\cal U$ of $X$ such that $H$ contains the Spanier group $π({\U},x)$ if and only if the core of $H$ in $π_1(X,x)$ is open in the quasitopological fundamental group $π_1^{qtop}(X,x)$ or equivalently it is open in the topological fundamental group $π_1^τ(X,x)$. As a consequence, using the relation between the Spanier groups and covering spaces, we give a classification for connected covering spaces of $X$ based on the conjugacy classes of subgroups with open core in $π_1^{qtop}(X,x)$. Finally, we give a necessary and sufficient condition for the existence of a semicovering. Moreover, we present a condition under which every semicovering of $X$ is a covering.

math.AT↗
arXiv2012

Spanier spaces and covering theory of non-homotopically path Hausdorff spaces

H. Fischer et al. (Topology and its Application, 158 (2011) 397-408.) introduced the Spanier group of a based space $(X,x)$ which is denoted by $\psp$. By a Spanier space we mean a space $X$ such that $\psp=π_1(X,x)$, for every $x\in X$. In this paper, first we give an example of Spanier spaces. Then we study the influence of the Spanier group on covering theory and introduce Spanier coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of Spanier coverings for non-homotopically path Hausdorff spaces. Finally, we study the topological properties of Spanier groups and find out a criteria for the Hausdorffness of topological fundamental groups.

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arXiv2012

Some outer commutator multipliers and capability of nilpotent products of cyclic groups

In this paper, first we obtain an explicit formula for an outer commutator multiplier of nilpotent products of cyclic groups with respect to the variety $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, $\mathfrak{N}_{c}M(\mathbb{Z}\st{n}* \mathbb{Z}\st{n}* ... \st{n}* \mathbb{Z}\st{n}* \mathbb{Z}_{r_1}\st{n}* \mathbb{Z}_{r_2}\st{n}* ... \st{n}* \mathbb{Z}_{r_t})$ where $r_{i+1}\mid r_i \ \ (1\leq i\leq t-1)$, $c_1+c_2+1\geq n$, $2c_2-c_1>2n-2$ and $(p,r_1)=1$ for all prime less than or equal $c_1+c_2+n$, second we give a necessary condition for these groups to be $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable.

math.GR↗
arXiv2012

On Hawaiian Groups of Some Topological Spaces

The paper is devoted to study the structure of Hawaiian groups of some topological spaces. We present some behaviors of Hawaiian groups with respect to product spaces, weak join spaces, cone spaces, covering spaces and locally trivial bundles. In particular, we determine the structure of the $n$-dimensional Hawaiian group of the $m$-dimensional Hawaiian earring space, for all $1\leq m\leq n$.

math.AT↗
arXiv2011

On Baer Invariants of Triples of Groups

In this paper, we develop the theory of Baer invariants for triples of groups. First, we focus on the general properties of the Baer invariant of triples. Second, we prove that the Baer invariant of a triple preserves direct limits of directed systems of triples of groups. Moreover, we present a structure for the nilpotent multiplier of a triple of the free product in some cases. Finally, we give some conditions in which the Baer invariant of a triple is a torsion group.

math.AT↗
arXiv2011

On Baer Invariants of Pairs of Groups

In this paper, we use the theory of simplicial groups to develop the Schur multiplier of a pair of groups $(G,N)$ to the Baer invariant of it, $\mathcal{V}M(G,N)$, with respect to an arbitrary variety $\mathcal{V}$. Moreover, we present among other things some behaviors of Baer invariants of a pair of groups with respect to the free product and the direct limit. Finally we prove that the nilpotent multiplier of a pair of groups does commute with the free product of finite groups of mutually coprime orders.

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arXiv2011

Polynilpotent Multipliers of some Nilpotent Products of Cyclic Groups II

This article is devoted to present an explicit formula for the $c$th nilpotent multiplier of nilpotent products of some cyclic groups $G={\bf {Z}}\stackrel{n_1}{*}{\bf {Z}}\stackrel{n_2}{*}...\stackrel{n_{t-1}}{*}{\bf {Z}}\stackrel{n_{t}}{*} {\bf {Z}}_{m_{t+1}}\stackrel{n_{t+1}}{*}{\bf {Z}}_{m_{t+2}}\stackrel{n_{t+2}}{*}...\stackrel{n_{k}}{*}{\bf{Z}}_{m_{k+1}}$, where $m_{i+1} | m_i$ for all $t+1 \leq i \leq k$ and $c \geq n_1\geq n_2\geq ...\geq n_t\geq ...\geq n_{k}$ such that $ (p,m_{t+1})=1$ for all prime $p \leq n_1$. Moreover, we compute the polynilpotent multiplier of the group $G$ with respect to the polynilpotent variety ${\mathcal N}_{c_1,c_2,...,c_s}$, where $c_1 \geq n_1.$

math.GR↗
arXiv2011

On Schur Multipliers of Pairs and Triples of Groups with Topological Approach

In this paper, using a relation between Schur multipliers of pairs and triples of groups, the fundamental group and homology groups of a homotopy pushout of Eilenberg-MacLane spaces, we present among other things some behaviors of Schur multipliers of pairs and triples with respect to free, amalgamated free, and direct products and also direct limits with topological approach.

math.AT↗
arXiv2011

A Remark on Generalized Covering Groups

Let ${\cal N}_c$ be the variety of nilpotent groups of class at most $c\ \ (c\geq 2)$ and $G=Z_r\oplus Z_s $ be the direct sum of two finite cyclic groups. It is shown that if the greatest common divisor of $r$ and $s$ is not one, then $G$ does not have any ${\cal N}_c$-covering group for every $c\geq 2$. This result gives an idea that Lemma 2 of J.Wiegold [6] and Haebich's Theorem [1], a vast generalization of the Wiegold's Theorem, can {\it not} be generalized to the variety of nilpotent groups of class at most $c\geq 2$.

math.GR↗
arXiv2011

Subgroup Theorems for the Baer-invariant of Groups

M.R.Jones and J.Wiegold in [3] have shown that if $G$ is a finite group with a subgroup $H$ of finite index $n$, then the $n$-th power of Schur multiplier of $G$, $M(G)^n$, is isomorphic to a subgroup of $M(H)$. In this paper we prove a similar result for the centre by centre by $w$ variety of groups, where $w$ is any outer commutator word. Then using a result of M.R.R.Moghaddam [6], we will be able to deduce a result of Schur's type (see [4,9]) with respect to the variety of nilpotent groups of class at most $c$ $(c\geq 1)$, when $c+1$ is any prime number or 4.

math.GR↗
arXiv2011

The Baer-invariant of a Semidirect Product

In 1972 K.I.Tahara [7,2 Theorem 2.2.5], using cohomological method, showed that if a finite group $G=T\rhd<N$ is the semidirect product of a normal subgroup $N$ and a subgroup $T$, then $M(T)$ is a direct factor of $M(G)$, where $M(G)$ is the Schur-multiplicator of $G$ and in the finite case, is the second cohomology group of $G$. In 1977 W.Haebich [1 Theorem 1.7] gave another proof using a different method for an arbitrary group $G$ . In this paper we generalize the above theorem . We will show that ${\cal N}_cM(T)$ is a direct factor of ${\cal N}_cM(G)$, where ${\cal N}_c$ [3 page 102] is the variety of nilpotent groups of class at most $c\geq 1$ and ${\cal N}_cM(G)$ is {\it the Baer-invariant} of the group $G$ with respect to the variety ${\cal N}_c$ [3 page 107] .

math.GR↗
arXiv2011

On the Order of Nilpotent Multipliers of Finite p-Groups

Let $G$ be a finite $p$-group of order $p^n$. YA. G. Berkovich (Journal of Algebra {\bf 144}, 269-272 (1991)) proved that $G$ is elementary abelian $p$-group if and only if the order of its Schur multiplier, $M(G)$, is at the maximum case. In this paper, first we find the upper bound $p^{χ_{c+1}{(n)}}$ for the order the $c$-nilpotent multiplier of $G$, $M^{(c)}(G)$, where $χ_{c+1}{(i)}$ is the number of basic commutators of weight $c+1$ on $i$ letters. Second, we obtain the structure of $G$, in abelian case, where $|M^{(c)}(G)|=p^{χ_{c+1}{(n-t)}}$, for all $0\leq t\leq n-1$. Finally, by putting a condition on the kernel of the left natural map of the generalized Stallings-Stammbach five term exact sequence, we show that an arbitrary finite $p$-group with the $c$-nilpotent multiplier of maximum order is an elementary abelian $p$-group.

math.GR↗
arXiv2011

Some Functorial Properties of Nilpotent Multipliers

In this paper, we are going to look at the $c$-nilpotent multiplier of a group $G$, ${\cal N}_cM(G)$, as a functor from the category of all groups, ${\cal G}roup$, to the category of all abelian groups, ${\cal A}b$, and focusing on some functional properties of it. In fact, by using some results of the first author and others and finding an explicit formula for the $c$-nilpotent multiplier of a finitely generated abelian group, we try to concentrate on the commutativity of the above functor with the two famous functors Ext and Tor.

math.GR↗
arXiv2011

On ${\cal N}_c$-Covering Groups of a Nilpotent Product of Cyclic Groups

The first author, in 2001, presented a structure for the Baer-invariant of a nilpotent product of cyclic groups with respect to the variety of nilpotent groups. In this paper, using the above structure, we prove the existence of an ${\cal N}_c$-covering group for a nilpotent product of a family of cyclic groups. We also present a structure for the ${\cal N}_c$-covering group.

math.GR↗
arXiv2011

On the Nilpotent Multiplier of a Free Product

In this paper, using a result of J. Burns and G. Ellis (Math. Z. 226 (1997) 405-28.), we prove that the $c$-nilpotent multiplier (the Baer-invariant with respect to the variety of nilpotent groups of class at most $c$, ${\cal N}_c$.) {\it does commute} with the free product of cyclic groups of mutually coprime order.

math.GR↗
arXiv2011

Some Notes on the Baer Invariant of a Nilpotent Product of Groups

W.Haebich (Bull. Austral. Math. Soc., 7, 1972, 279-296) presented a formula for the Schur multiplier of a regular product of groups. In this paper first, it is shown that the Baer-invariant of a nilpotent product of groups with respect to the variety of nilpotent groups has a homomorphic image and in finite case a subgroup of Haebich's type. Second a formula will be presented for the Baer-invariant of a nilpotent product of cyclic groups with respect to the variety of nilpotent groups.

math.GR↗