arXiv subjects

Mehmet Akif Erdal

Publications and source records attributed to Mehmet Akif Erdal.

6 recordsLinked to original sources

arXiv2026

Identifying domatic partitions via graph dynamical systems

For a graph $G=(V,E),$ a set of vertices $D\subseteq V $ is called a dominating set if every vertex in $V\backslash D$ is adjacent to a vertex in $D.$ A domatic-$2$-partition of $G$ is a partition of its vertices into two disjoint dominating sets. In this paper, for a finite simple connected graph $G,$ we construct a graph dynamical system $F$ and show that the set of dominating sets of $G$ are in one-to-one correspondence with the image of the action map of $F$. Moreover, we obtain the set of all domatic-$2$-partitions of $G$ from the set of all periodic orbits of $F.$ Finally, we extended actions of two dynamical systems to an action of a free semigroup on two letters, and determine independent dominating sets and idomatic partitions using its maximal invariant subset with a reversible action.

math.CO↗

arXiv2023

Homotopy theory of monoid actions via group actions and an Elmendorf style theorem

Let $M$ be a monoid and $G:\mathbf{Mon} \to \mathbf{Grp}$ be the group completion functor from monoids to groups. Given a collection $\mathcal{X}$ of submonoids of $M$ and for each $N\in \mathcal{X}$ a collection $\mathcal{Y}_N$ of subgroups of $G(N)$, we construct a model structure on the category of $M$-spaces and $M$-equivariant maps, called the $(\mathcal X,\mathcal Y)$-model structure, in which weak equivalences and fibrations are induced from the standard $\mathcal{Y}_N$-model structures on $G(N)$-spaces for all $N\in \mathcal{X}$. We also show that for a pair of collections $(\mathcal{X},\mathcal{Y})$ there is a small category ${\mathbf O}_{(\mathcal{X},\mathcal{Y})}$ whose objects are $M$-spaces $M\times_NG(N)/H$ for each $N\in \mathcal X$ and $H\in \mathcal Y_N$ and morphisms are $M$-equivariant maps, such that the $(\mathcal X,\mathcal Y)$-model structure on the category of $M$-spaces is Quillen equivalent to the projective model structure on the category of contravariant ${\mathbf O}_{(\mathcal{X},\mathcal{Y})}$-diagrams of spaces.

math.CT↗

arXiv2021

Stabilization and costabilization with respect to an action of a monoidal category

We study actions of monoidal categories on objects in a suitably enriched $2$-category, and applications in stable homotopy theory. Given a monoidal category $\mathcal{I}$ and an $\mathcal{I}$-object $\mathcal{A}$, the (co)stabilization of $\mathcal{A}$ is obtained by universally forcing the $\mathcal{I}$-action to be reversible so that every object of $\mathcal{I}$ acts on $\mathcal{A}$ by auto-equivalences. We introduce a notion of $\mathcal{I}$-equivariance for morphisms between $\mathcal{I}$-objects and give constructions of stabilization and costabilization in terms of weak ends and coends in an enriched $2$-category of $\mathcal{I}$-objects and $\mathcal{I}$-equivariant morphisms. We observe that the stabilization of a relative category with respect to an action coincides with the usual notion of stabilization in stable homotopy theory when the action is defined by loop space functors. We show that several examples that exist in the literature, including various categories of spectra, fit into our setting after fixing $\mathcal{A}$ and the $\mathcal{I}$-action on it. In particular, categories of sequential spectra, coordinate free spectra, genuine equivariant spectra, parameterized spectra indexed by vector bundles are obtained in terms of weak ends in the $2$-category of relative categories. On the other hand, the costabilization of a relative category with respect to an action gives a stable relative category akin to a version Spanier-Whitehead category. In particular, we establish a form of duality between constructions of stable homotopy categories by spectra and by Spanier-Whitehead like categories.

math.CT↗

arXiv2019

A model structure via orbit spaces for equivariant homotopy

For a given group $G$ and a collection of subgroups $\mathcal F$ of $G$, we show that there exist a left induced model structure on the category of right $G$-simplicial sets, in which the weak equivalences and cofibrations are the maps that induce weak equivalences and cofibrations on the $H$-orbits for all $H$ in $\mathcal F$. This gives a model categorical criterion for maps that induce weak equivalences on $H$-orbits to be weak equivalences in the $\mathcal F$-model structure.

math.AT↗

arXiv2018

Semigroup actions on sets and the Burnside ring

In this paper we discuss some enlargements of the category of sets with semigroup actions and equivariant functions. We show that these enlarged categories possess two idempotent endofunctors. In the case of groups these enlarged categories are equivalent to the usual category of group actions and equivariant functions, and these idempotent endofunctors reverse a given action. For a general semigroup we show that these enlarged categories admit homotopical category structures defined by using these endofunctors and show that up to homotopy these categories are equivalent to the usual category of sets with semigroup actions. We finally construct the Burnside ring of a monoid by using homotopical structure of these categories, so that when the monoid is a group this definition agrees with the usual definition, and we show that when the monoid is commutative, its Burnside ring is equivalent to the Burnside ring of its Gröthendieck group.

math.AT↗

arXiv2018

On smooth manifolds with homotopy type of a homology sphere

In this paper we study $\mathcal M(X)$, the set of diffeomorphism classes of smooth manifolds with the simple homotopy type of $X$, via a map $Ψ$ from $\mathcal M(X)$ into the quotient of $K(X)=[X,BSO]$ by the action of the group of homotopy classes of simple self equivalences of $X$. The map $Ψ$ describes which bundles over $X$ can occur as normal bundles of manifolds in $\mathcal M(X)$. We determine the image of $Ψ$ when $X$ belongs to a certain class of homology spheres. In particular, we find conditions on elements of $K(X)$ that guarantee they are pullbacks of normal bundles of manifolds in $\mathcal M(X)$.

math.AT↗