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M. Firat Arikan

Publications and source records attributed to M. Firat Arikan.

14 recordsLinked to original sources
arXiv2025

Tight Contact Structures on Contact Mapping Tori and their Folded Sums

It is known that the folded sum of two contact mapping tori whose fibers are compact exact symplectic manifolds having a common convex boundary (called the ``fold'') admits a cooriented contact structure compatible with the obvious fibration map onto the circle. Here we first provide an alternative bundle-theoretical construction of such a ``folded'' contact structure based on a gluing process near the fold. Moreover, we prove that in any odd dimension $2n+1\geq 7$ a folded contact structure on a folded sum of two contact mapping tori is tight if the induced contact form on the (common) contact fold admits no contractible Reeb orbit. In particular, any contact mapping torus of an odd dimension $2n+1\geq 7$ is tight if the induced contact form on the convex boundary of a fiber admits no contractible Reeb orbit.

math.GT↗
arXiv2024

Compatible Relative Lefschetz Fibrations On Admissible Relative Stein Pairs

For more than two decades it has been known that any compact Stein surface (of real dimension four) admits a compatible Lefschetz fibration over a two-disk. More recently, Giroux and Pardon have generalized this result by giving a complex geometric proof for the existence of compatible Lefschetz fibrations on Stein domains of any even dimension. As a preparatory step in proving the former, Akbulut and Ozbagci have shown that there exist infinitely many pairwise non-equivalent Lefschetz fibrations on the four-ball by using a result of Lyon constructing fibrations on the complements of (p,q)-torus links in the three-sphere. In this paper, we first extend this result to obtain compatible Lefschetz fibrations on the six-ball whose pages are (p, q, 2)-Brieskorn varieties, and then construct a compatible 'relative' Lefschetz fibrations on any Stein domain (of dimension six) which admit a certain ('admissible') 'relative Stein pair' structure. In particular, we provide a purely topological proof for the existence of Lefschetz fibrations on specific 6-dimensional Stein domains.

math.GT↗
arXiv2023

Compatible Relative Open Books on Relative Contact Pairs via Generalized Square Bridge Diagrams

Using square bridge position, Akbulut-Ozbagci and later Arikan gave algorithms both of which construct an explicit compatible open book decomposition on a closed contact $3$-manifold which results from a contact $(\pm 1)$-surgery on a Legendrian link in the standard contact $3$-sphere. In this article, we introduce the ``generalized square bridge position'' for a Legendrian link in the standard contact $5$-sphere and partially generalize this result to the dimension five via an algorithm which constructs relative open book decompositions on relative contact pairs.

math.GT↗
arXiv2021

Maximal Page Crossing Numbers of Legendrian Surfaces in Closed Contact 5-Manifolds

We introduce a new Legendrian isotopy invariant for any closed orientable Legendrian surface $L$ embedded in a closed contact $5$-manifold $(M, ξ)$ which admits an "admissable" open book $(B, f)$ (supporting $ξ$) for $L$. We show that to any such $L$ and a fixed page $X$, one can assign an integer $M\mathcal{P}_{X}(L)$, called "Relative Maximal Page Crossing Number of $L$ with respect to $X$", which is invariant under Legendrian isotopies of $L$. We also show that one can extend this to a page-free invariant, i.e., one can assign an integer $M\mathcal{P}_{(B,f)}(L)$, called "Absolute Maximal Page Crossing Number of $L$ with respect to $(B, f)$", which is invariant under Legendrian isotopies of $L$. In particular, this new invariant distinguishes Legendrian surfaces in the standard five-sphere which can not be distinguished by Thurston-Bennequin invariant.

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arXiv2018

Lefschetz Fibrations on Compact Stein Manifolds

Here we prove that up to diffeomorphism every compact Stein manifold W of dimension 2n+2>4 admits a Lefschetz fibration over the two-disk with Stein regular fibers, such that the monodromy of the fibration is a symplectomorphism induced by compositions of right-handed Dehn twists along embedded Lagrangian n-spheres on the generic fiber. This generalizes the Stein surface case of n=1, previously proven by Loi-Piergallini and Akbulut-Ozbagci. More precisely, we show that up to Liouville isomorphism any Weinstein domain W admits a compatible compact convex Lefschetz fibration with Weinstein regular fibers and with the same monodromy description stated above. Moreover, the induced convex open book supports the induced contact structure on the boundary of W.

math.GT↗
arXiv2018

On Legendrian Embbeddings into Open Book Decompositions

We study Legendrian embeddings of a compact Legendrian submanifold $L$ sitting in a closed contact manifold $(M,ξ)$ whose contact structure is supported by a (contact) open book $\mathcal{OB}$ on $M$. We prove that if $\mathcal{OB}$ has Weinstein pages, then there exist a contact structure $ξ'$ on $M$, isotopic to $ξ$ and supported by $\mathcal{OB}$, and a contactomorphism $f:(M,ξ) \to (M,ξ')$ such that the image $f(L)$ of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of $\mathcal{OB}$.

math.SG↗
arXiv2014

Contact Symplectic Fibrations and Fiber Connected Sum

We consider certain type of fiber bundles with odd dimensional compact contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations "contact symplectic fibrations". By a result of Hajduk-Walczak, some of these admit contact structures which are "compatible" (in a certain sense) with the corresponding fibration structures. We show that this result can be extended to get a compatible contact structure on any contact symplectic fibration, and also that isotopic contact structures on the base produce isotopic contact structures on the total space.. Moreover, we prove that the fiber connected summing of two contact symplectic fibrations along their fibers results in another contact symplectic fibration which admits a compatible contact structure agreeing with the original ones away from the region where we take fiber connected sum, and whose base is the contact connected sum of the original contact bases.

math.GT↗
arXiv2013

Legendrian Realization in Convex Lefschetz Fibrations and Convex Stabilizations

In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension $2n+2$ which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on $\partial W$, which we call \emph{convex open book}, induced by a compact convex Lefschetz fibration on $W$ carries the contact structure induced by the convex symplectic structure (i.e., Liouville structure) on $W$. Here we show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on $W$, any simply connected embedded Lagrangian submanifold of a page in a convex open book on $\partial W$ can be assumed to be Legendrian in $\partial W$ with the induced contact structure. This can be thought as the extension of Giroux's Legendrian realization (which holds for contact open books) for the case of convex open books. Moreover, a result of Akbulut-Arikan implies that there is a one-to-one correspondence between convex stabilizations of a convex open book and convex stabilizations of the corresponding compact convex Lefschetz fibration. We also show that the convex stabilization of a compact convex Lefschetz fibration on $W$ yields a compact convex Lefschetz fibration on a Liouville domain $W'$ which is exact symplectomorphic to a \emph{positive expansion} of $W$. In particular, with the induced structures $\partial W$ and $\partial W'$ are contactomorphic.

math.SG↗
arXiv2012

Contact Structures on G_2-Manifolds and Spin 7-Manifolds

We show that there exist infinitely many pairwise distinct non-closed G_2-manifolds (some of which have holonomy full G_2) such that they admit co-oriented contact structures and have co-oriented contact submanifolds which are also associative. Along the way, we prove that there exists a tubular neighborhood N of every orientable three-submanifold Y of an orientable seven-manifold with spin structure such that for every co-oriented contact structure on Y, N admits a co-oriented contact structure such that Y is a contact submanifold of N. Moreover, we construct infinitely many pairwise distinct non-closed seven-manifolds with spin structures which admit co-oriented contact structures and retract onto co-oriented contact submanifolds of co-dimension four.

math.DG↗
arXiv2012

Stabilizations via Lefschetz Fibrations and Exact Open Books

We show that if a contact open book $(Σ,h)$ on a $(2n+1)$-manifold $M$ ($n\geq1$) is induced by a Lefschetz fibration $π:W \to D^2$, then there is a one-to-one correspondence between positive stabilizations of $(Σ,h)$ and \emph{positive stabilizations} of $π$. More precisely, any positive stabilization of $(Σ,h)$ is induced by the corresponding positive stabilization of $π$, and conversely any positive stabilization of $π$ induces the corresponding positive stabilization of $(Σ,h)$. We define \emph{exact open books} as boundary open books of compatible exact Lefschetz fibrations, and show that any exact open book carries a contact structure. Moreover, we prove that there is a one-to-one correspondence (similar to the one above) between \emph{convex stabilizations} of an exact open book and \emph{convex stabilizations} of the corresponding compatible exact Lefschetz fibration. We also show that convex stabilization of compatible exact Lefschetz fibrations produces symplectomorphic completions.

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arXiv2011

Existence of Compatible Contact Structures on $G_2$-manifolds

In this paper, we show the existence of (co-oriented) contact structures on certain classes of $G_2$-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with $G_2$-structure) admits an almost contact structure. We also construct explicit almost contact metric structures on manifolds with $G_2$-structures.

math.DG↗
arXiv2011

On the Classification of Planar Contact Structures

In this paper, we focus on contact structures supported by planar open book decompositions. We study right-veering diffeomorphisms to keep track of overtwistedness property of contact structures under some monodromy changes. As an application we give infinitely many examples of overtwisted contact structures supported by open books whose pages are the four-punctured sphere, and also we prove that a certain family is holomorphically fillable using lantern relation.

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