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R. Inanc Baykur

Publications and source records attributed to R. Inanc Baykur.

At least 19 recordsLinked to original sources
arXiv2026

Exotic knottings and symmetries of surfaces in 4-manifolds

We study exotic knottings of surfaces in 4-manifolds through their ambient symmetries. We first give a general recipe for producing projectively rigid surfaces, for which every smoothly extendable self-diffeomorphism acts on first homology by plus or minus the identity. For every integer g >0, a refinement of this construction yields a finite sequence of genus-g surfaces F_0, ..., F_2g contained in a 4-manifold X_g. These surfaces are topologically isotopic and topologically flexible: every orientation-preserving self-diffeomorphism of F_i can be realized by a self-homeomorphism of X_g preserving F_i. Successive knotting, however, rules out increasingly many projective homological symmetries, revealing a finer knottedness phenomenon. The first two constructions combine iterated rim surgery with the convex geometry of Newton polytopes of relative Seiberg-Witten invariants. We also use hyperbolic geometry to construct a totally geodesic surface of positive genus whose smooth and topological extendable mapping class groups are both trivial.

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arXiv2026

Planar contact 3-manifolds with infinitely many Stein fillings

We prove that there are infinitely many closed contact 3-manifolds supported by planar open books, each admitting infinitely many pairwise non-homeomorphic Stein fillings. This answers K3 Problem 4.105. As a corollary, there are contact 3-manifolds that admit infinitely many Stein fillings but do not admit arbitrarily large ones.

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arXiv2025

Stein fillings vs. Milnor fibers

Given a link of a normal surface singularity with its canonical contact structure, we compare the collection of its Stein fillings to its Milnor fillings (that is, Milnor fibers of possible smoothings). We prove that, unlike Stein fillings, Milnor fillings of a given link have bounded topology; for links of sandwiched singularities, we further establish that there are only finitely many Milnor fillings. We discuss some other obstructions for a Stein filling to be represented by a Milnor fiber, and for various types of singularities, including simple classes like cusps and triangle singularities, we produce Stein fillings that do not come from Milnor fibers or resolutions.

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arXiv2024

Smooth structures on four-manifolds with finite cyclic fundamental groups

For each nonnegative integer m we show that any closed, oriented topological four-manifold with fundamental group Z_{4m+2} and odd intersection form, with possibly seven exceptions, either admits no smooth structure or admits infinitely many distinct smooth structures up to diffeomorphism. Moreover, we construct infinite families of non-complex irreducible fake projective planes with diverse fundamental groups.

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arXiv2024

Exotic 4-manifolds with signature zero

We produce infinitely many distinct irreducible smooth 4-manifolds homeomorphic to #(2m+1)(CP^2 # -CP^2) and #(2n+1)(S^2 x S^2), respectively, for each m>3 and n>4. These provide the smallest exotic closed simply connected 4-manifolds with signature zero known to date, and in each one of these homeomorphism classes, we get minimal symplectic 4-manifolds. Our novel exotic 4-manifolds are derived from fairly special small Lefschetz fibrations we build via positive factorizations in the mapping class group, with spin and non-spin monodromies.

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arXiv2024

On four-manifolds without 1- and 3-handles

We note that infinitely many irreducible, closed, simply connected 4-manifolds, with prescribed signature and spin type, admit perfect Morse functions, i.e. they can be given handle decompositions without 1- and 3-handles. In particular, there are many such 4-manifolds homeomorphic but not diffeomorphic to the standard 4-manifolds # m (S^2 x S^2) and # n (CP^2 # -CP^2), respectively, which answers Problem 4.91 on Kirby's 1997 list.

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arXiv2023

Nielsen realization in dimension four and projective twists

We demonstrate the existence of numerous non-spin 4-manifolds for which the smooth Nielsen realization problem fails; namely, there exist finite subgroups of their mapping class groups that cannot be realized by any group of diffeomorphisms. This extends and complements recent results for spin 4-manifolds. Our examples span virtually all possible intersection forms, both even and odd, indefinite and definite, and include many irreducible 4-manifolds. To derive these examples, we study multi-twists, projective twists, and multi-reflections, which are all mapping classes supported around collections of embedded spheres and projective planes. Our obstructions to Nielsen realization are based on the work of Konno. We investigate projective twists in further detail, and notably, employ them to show that, for many closed symplectic 4-manifolds, the symplectic Torelli group is not generated by squared Dehn twists.

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arXiv2023

Spin Lefschetz fibrations are abundant

We prove that any finitely presented group can be realized as the fundamental group of a spin Lefschetz fibration over the 2-sphere. We moreover show that any admissible lattice point in the symplectic geography plane below the Noether line can be realized by a simply-connected spin Lefschetz fibration.

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arXiv2023

Geography of surface bundles over surfaces

We construct symplectic surface bundles over surfaces with positive signatures for all but 18 possible pairs of fiber and base genera. Meanwhile, we determine the commutator lengths of a few new mapping classes.

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arXiv2022

Geography of symplectic Lefschetz fibrations and rational blowdowns

We produce simply connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides a symplectic extension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic $4$-manifolds. We present an example of a construction of a minimal symplectic exotic $\mathbb{CP} \# 5\,\overline{\mathbb{CP}}$ through this procedure applied to a genus-$3$ fibration.

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arXiv2021

Branched covering simply-connected 4-manifolds

We prove that any closed simply-connected smooth 4-manifold is 16-fold branched covered by a product of an orientable surface with the 2-torus, where the construction is natural with respect to spin structures. In particular this solves Problem 4.113(C) in Kirby's list. We also discuss analogous results for other families of 4-manifolds with infinite fundamental groups.

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arXiv2020

Lefschetz fibrations with arbitrary signature

We develop techniques to construct explicit symplectic Lefschetz fibrations over the 2-sphere with any prescribed signature and any spin type when the signature is divisible by 16. This solves a long-standing conjecture on the existence of such fibrations with positive signature. As applications, we produce symplectic 4-manifolds that are homeomorphic but not diffeomorphic to connected sums of S^2 x S^2, with the smallest topology known to date, as well as larger examples as symplectic Lefschetz fibrations.

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arXiv2019

Unchaining surgery and topology of symplectic 4-manifolds

We study a symplectic surgery operation we call unchaining, which effectively reduces the second Betti number and the symplectic Kodaira dimension at the same time. Using unchaining, we give novel constructions of symplectic Calabi-Yau surfaces from complex surfaces of general type, as well as from rational and ruled surfaces via the natural inverse of this operation. Combining the unchaining surgery with others, which all correspond to certain monodromy substitutions for Lefschetz pencils, we provide further applications, such as a complete resolution of a conjecture of Stipsicz on the existence of exceptional sections in Lefschetz fibrations, new constructions of exotic symplectic 4-manifolds, and inequivalent pencils of the same genera and the same number of base points on families of symplectic 4-manifolds. Meanwhile, we give a handy criterion for determining from the monodromy of a pencil whether its total space is spin or not.

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arXiv2018

Inequivalent Lefschetz fibrations on rational and ruled surfaces

In this short note, we give an explicit construction of inequivalent Lefschetz pencils and fibrations of same genera on blow-ups of all rational and ruled surfaces. This complements our earlier results, concluding that every symplectic 4-manifold, after sufficiently many blow-ups, admits inequivalent Lefschetz pencils and fibrations, which cannot be obtained from one another even via any sequence of fibered Luttinger surgeries.

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arXiv2018

Dissolving knot surgered 4-manifolds by classical cobordism arguments

The purpose of this note is to show that classical cobordism arguments, which go back to the pioneering works of Mandelbaum and Moishezon, provide quick and unified proofs of any knot surgered compact simply-connected 4-manifold X_K becoming diffeomorphic to X after a single stabilization by connected summing with S^2 x S^2 or CP^2 # -CP^2, and almost complete decomposability of X_K for many almost completely decomposable X, such as the elliptic surfaces.

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arXiv2017

Simplified broken Lefschetz fibrations and trisections of 4-manifolds

Shapes of four dimensional spaces can be studied effectively via maps to standard surfaces. We explain, and illustrate by quintessential examples, how to simplify such generic maps on 4-manifolds topologically, in order to derive simple decompositions into much better understood manifold pieces. Our methods not only allow us to produce various interesting families of examples, but also to establish a correspondence between simplified broken Lefschetz fibrations and simplified trisections of closed, oriented 4-manifolds.

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arXiv2017

Simplifying indefinite fibrations on 4-manifolds

We present explicit algorithms for simplifying the topology of indefinite fibrations on 4-manifolds, which include broken Lefschetz fibrations and indefinite Morse 2-functions. The algorithms consist of sequences of moves, which modify indefinite fibrations in smooth 1-parameter families. In particular, given an arbitrary broken Lefschetz fibration, we show how to turn it to one with directed and embedded round (indefinite fold) image, and to one with all the fibers and the round locus connected. We also show how to realize any given null-homologous 1-dimensional submanifold with prescribed local models for its components as the round locus of such a broken Lefschetz fibration. These algorithms allow us to give purely topological and constructive proofs of the existence of simplified broken Lefschetz fibrations and Morse 2-functions on general 4-manifolds, and a theorem of Auroux-Donaldson-Katzarkov on the existence of broken Lefschetz pencils with directed embedded round image on near-symplectic 4-manifolds. We moreover establish a correspondence between broken Lefschetz fibrations and Gay-Kirby trisections of 4-manifolds, and show the existence of simplified trisections on all 4-manifolds. Building on this correspondence, we provide several new constructions of trisections, including infinite families of genus-3 trisections with homotopy inequivalent total spaces, and exotic same genera trisections of 4-manifolds in the homeomorphism classes of complex rational surfaces.

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