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Burak Ozbagci

Publications and source records attributed to Burak Ozbagci.

At least 19 recordsLinked to original sources
arXiv2026

The Gompf $θ$-Invariant of Canonical Contact Structures via Legendrian Surgery

Let $Γ$ be a minimal connected negative-definite plumbing tree with all vertices of genus zero, and let $Y_Γ$ be the oriented link of the corresponding normal complex surface singularity, equipped with its canonical contact structure $ξ_{\rm can}$. We give an explicit Legendrian surgery description of $ξ_{\rm can}$, showing that it is the unique consistent diagram-realizable contact structure on $Y_Γ$, up to isomorphism. We then derive a closed-form formula for Gompf's $θ$-invariant of $ξ_{\rm can}$ in the Seifert fibered case, expressed purely in terms of the Hirzebruch--Jung continued fraction expansions of the normalized Seifert invariants, and prove a recursive leaf-to-root formula for arbitrary plumbing trees. The Seifert formula recovers previously known formulas for lens spaces, dihedral manifolds, and small Seifert fibered spaces with complementary legs, and agrees with the Némethi--Nicolaescu expression via the classical Hirzebruch--Zagier identity. As a final application we show that $ξ_{\rm can}$ strictly minimizes $θ$ among all diagram-realizable contact structures on $Y_Γ$, and we use this to rule out symplectic rational homology ball fillings for a large class of Stein fillable contact rational homology $3$-spheres.

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arXiv2026

Complementary legs and symplectic rational balls

We show that a small Seifert fibered space with complementary legs does not symplectically bound a rational homology ball for at least one choice of orientation. In the case $e_0\leq -1$, we characterize when a small Seifert fibered space with uniquely complementary legs symplectically bounds a rational homology ball. In the case $e_0\geq 0$, we characterize when a small Seifert fibered space with complementary legs, equipped with a balanced contact structure, symplectically bounds a rational homology ball. Our results highlight a sharp contrast with the smooth category, where many more such Seifert fibered spaces are known to bound smooth rational homology balls. As a consequence of the results above, we also complete the classification of contact structures on oriented spherical $3$-manifolds that admit symplectic rational homology ball fillings. In particular, we show that a closed, oriented $3$-manifold with finite fundamental group admits at most six contact structures, up to isotopy, which are symplectically fillable by rational homology balls.

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arXiv2026

Unbraided wiring diagrams for Stein fillings of lens spaces

In a previous work, we constructed a planar Lefschetz fibration on each Stein filling of any lens space equipped with its canonical contact structure. Here we describe an algorithm to draw an unbraided wiring diagram whose associated planar Lefschetz fibration obtained by the method of Plamenevskaya and Starkston, where the lens space with its canonical contact structure is viewed as the contact link of a cyclic quotient singularity, is equivalent to the Lefschetz fibration we constructed on each Stein filling of the lens space at hand. Coupled with the work of Plamenevskaya and Starkston, we obtain the following result as a corollary: The wiring diagram we describe can be extended to an arrangement of symplectic graphical disks in $\mathbb{C}^2$ with marked points, including all the intersection points of these disks, so that by removing the proper transforms of these disks from the blowup of $\mathbb{C}^2$ along those marked points one recovers the Stein filling along with the Lefschetz fibration. Moreover, the arrangement is related to the decorated plane curve germ representing the cyclic quotient singularity by a smooth graphical homotopy. As another application, we set up an explicit bijection between the Stein fillings of any lens space with its canonical contact structure, and the Milnor fibers of the corresponding cyclic quotient singularity, which was first obtained by Némethi and Popescu-Pampu, using different methods.

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arXiv2025

Symplectic rational homology ball fillings of Seifert fibered spaces

We characterize when some small Seifert fibered spaces can be the convex boundaries of symplectic rational homology balls and give strong restrictions for others to bound such manifolds. In particular, we show that the only spherical $3$-manifolds, oriented as links of the corresponding quotient singularities, which admit symplectic rational homology ball fillings, are the lens spaces $L(p^2,pq-1)$ previously identified by Lisca. In a different direction, we provide evidence for Gompf's conjecture that Brieskorn spheres do not bound Stein domains in $\mathbb{C}^2$. Finally, we establish restrictions on Lagrangian disk fillings of certain Legendrian knots in small Seifert fibered spaces.

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arXiv2024

A characterization of quasipositive two-bridge knots

We prove a simple necessary and sufficient condition for a two-bridge knot K(p,q) to be quasipositive, based on the continued fraction expansion of p/q. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in the Appendix.

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arXiv2023

Complex vs convex Morse functions and geodesic open books

Suppose that $Σ$ is a closed and oriented surface equipped with a Riemannian metric. In the literature, there are three seemingly distinct constructions of open books on the unit (co)tangent bundle of $Σ$, having complex, contact, and dynamical flavors, respectively. Each one of these constructions is based on either an admissible divide or an ordered Morse function on $Σ$. We show that the resulting open books are pairwise isotopic provided that the ordered Morse function is adapted to the admissible divide on $Σ$. Moreover, we observe that if $Σ$ has positive genus, then none of these open books are planar and furthermore, we determine the only cases when they have genus one pages.

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arXiv2022

Generalizations of planar contact manifolds to higher dimensions

Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions. We study some basic topological properties of iterated planar contact manifolds and discuss several examples and constructions showing that many contact manifolds are iterated planar. We also observe that for any odd integer m > 3, any finitely presented group can be realized as the fundamental group of some iterated planar contact manifold of dimension m. Moreover, we introduce another generalization of three dimensional planar contact manifolds that we call projective. Finally, building symplectic cobordisms via open books, we show that some projective contact manifolds admit explicit symplectic caps.

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arXiv2022

Rational blowdown graphs for symplectic fillings of lens spaces

In a previous work, we proved that each minimal symplectic filling of any oriented lens space, viewed as the singularity link of some cyclic quotient singularity and equipped with its canonical contact structure, can be obtained from the minimal resolution of the singularity by a sequence of symplectic rational blowdowns along linear plumbing graphs. Here we give a dramatically simpler visual presentation of our rational blowdown algorithm in terms of the triangulations of a convex polygon. As a consequence, we are able to organize the symplectic deformation equivalence classes of all minimal symplectic fillings of any given lens space equipped with its canonical contact structure, as a graded, directed, rooted, and connected graph, where the root is the minimal resolution of the corresponding cyclic quotient singularity and each directed edge is a symplectic rational blowdown along an explicit linear plumbing graph. Moreover, we provide an upper bound for the rational blowdown depth of each minimal symplectic filling.

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arXiv2021

Lefschetz fibrations on nonorientable 4-manifolds

Let $W$ be a nonorientable $4$-dimensional handlebody without $3$- and $4$-handles. We show that $W$ admits a Lefschetz fibration over the $2$-disk, whose regular fiber is a nonorientable surface with nonempty boundary. This is an analogue of a result of Harer obtained in the orientable case. As a corollary, we obtain a $4$-dimensional proof of the fact that every nonorientable closed $3$-manifold admits an open book decomposition, which was first proved by Berstein and Edmonds using branched coverings. Moreover, the monodromy of the open book we obtain for a given $3$-manifold belongs to the twist subgroup of the mapping class group of the page. In particular, we construct an explicit minimal open book for the connected sum of arbitrarily many copies of the product of the circle with the real projective plane. We also obtain a relative trisection diagram for $W$, based on the nonorientable Lefschetz fibration we construct, similar to the orientable case first studied by Castro. As a corollary, we get trisection diagrams for some closed $4$-manifolds, e.g. the product of the $2$-sphere with the real projective plane, by doubling $W$. Moreover, if $X$ is a closed nonorientable $4$-manifold which admits a Lefschetz fibration over the $2$-sphere, equipped with a section of square $\pm 1$, then we construct a trisection diagram of $X$, which is determined by the vanishing cycles of the Lefschetz fibration. Finally, we include some simple observations about low-genus Lefschetz fibrations on closed nonorientable $4$-manifolds.

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arXiv2020

On open books for nonorientable 3-manifolds

We show that the monodromy of Klassen's genus two open book for $P^2 \times S^1$ is the $Y$-homeomorphism of Lickorish, which is also known as the crosscap slide. Similarly, we show that $S^2 \widetilde{\times} S^1$ admits a genus two open book whose monodromy is the crosscap transposition. Moreover, we show that each of $P^2 \times S^1$ and $S^2 \widetilde{\times} S^1$ admits infinitely many isomorphic genus two open books whose monodromies are mutually nonisotopic. Furthermore, we include a simple observation about the stable equivalence classes of open books for $P^2 \times S^1$ and $S^2 \widetilde{\times} S^1$. Finally, we formulate a version of Gabai's theorem about the Murasugi sum of open books, without imposing any orientability assumption on the pages.

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arXiv2019

Stein and Weinstein structures on disk cotangent bundles of surfaces

In 1998, Gompf described a Stein domain structure on the disk cotangent bundle of any closed surface S, by a Legendrian handlebody diagram. We prove that Gompf's Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorphic to the unit cotangent bundle of S equipped with its canonical contact structure. As a corollary, we obtain a surgery diagram for the canonical contact structure on the unit cotangent bundle of S.

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arXiv2017

Trisections of 4-manifolds via Lefschetz fibrations

We develop a technique for gluing relative trisection diagrams of $4$-manifolds with nonempty connected boundary to obtain trisection diagrams for closed $4$-manifolds. As an application, we describe a trisection of any closed $4$-manifold which admits a Lefschetz fibration over $S^2$ equipped with a section of square $-1$, by an explicit diagram determined by the vanishing cycles of the Lefschetz fibration. In particular, we obtain a trisection diagram for some simply connected minimal complex surface of general type. As a consequence, we obtain explicit trisection diagrams for a pair of closed $4$-manifolds which are homeomorphic but not diffeomorphic. Moreover, we describe a trisection for any oriented $S^2$-bundle over any closed surface and in particular we draw the corresponding diagrams for $T^2 \times S^2$ and $T^2 \tilde{\times} S^2$ using our gluing technique. Furthermore, we provide an alternate proof of a recent result of Gay and Kirby which says that every closed $4$-manifold admits a trisection. The key feature of our proof is that Cerf theory takes a back seat to contact geometry.

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arXiv2016

Fillings of unit cotangent bundles of nonorientable surfaces

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism.

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arXiv2016

Generalized plumbings and Murasugi sums

We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai's credo "the Murasugi sum is a natural geometric operation" holds. In particular, we prove that the sum of the pages of two open books is again a page of an open book and that there is an associated summing operation of Morse maps. We conclude with several open questions relating this work with singularity theory or contact topology.

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arXiv2015

Contact open books with exotic pages

We consider a fixed contact 3-manifold that admits infinitely many compact Stein fillings which are all homeomorphic but pairwise non-diffeomorphic. Each of these fillings gives rise to a closed contact 5-manifold described as a contact open book whose page is the filling at hand and whose monodromy is the identity symplectomorphism. We show that the resulting infinitely many contact 5-manifolds are all diffeomorphic but pairwise non-contactomorphic. Moreover, we explicitly determine these contact 5-manifolds.

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