R. Inanc Baykur

Brandeis University, Волтам, Massachusetts, United States

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Publications (25)10.2 Total impact

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    R. Inanc Baykur, Nathan Sunukjian
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    ABSTRACT: In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic but not smoothly isotopic. We prove that any pair of embedded surfaces in the same homology class become smoothly isotopic after stabilizing them by handle additions in the ambient 4-manifold, which can moreover assumed to be attached in a standard way (locally and unknottedly) in many favorable situations. In particular, any exotically knotted pair of surfaces with cyclic fundamental group complements become smoothly isotopic after a same number of standard stabilizations - analogous to C.T.C. Wall's celebrated result on the stable equivalence of simply-connected 4-manifolds. We moreover show that all constructions of exotic knottings of surfaces we are aware of, which display a good variety of techniques and ideas, produce surfaces that become smoothly isotopic after a single stabilization.
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    R. Inanc Baykur, Kenta Hayano
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    ABSTRACT: In this article, we characterize isomorphism classes of Lefschetz fibrations with multisections via their monodromy factorizations. We prove that two Lefschetz fibrations with multisections are isomorphic if and only if their monodromy factorizations in the relevant mapping class groups are related to each other by a finite collection of modifications, which extend the well-known Hurwitz equivalence. This in particular characterizes isomorphism classes of Lefschetz pencils. We then show that, from simple relations in the mapping class groups, one can derive new (and old) examples of Lefschetz fibrations which cannot be written as fiber sums of blown-up pencils.
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    ABSTRACT: We study which surface mapping classes can be factorized into arbitrarily large number of positive Dehn twists or only of a fixed number. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds / Stein fillings of contact 3-manifolds coming from the topology of Lefschetz pencils / open books on them, we completely determine when a boundary multitwist admits arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements with such factorizations. In particular, we observe that only symplectic 4-manifolds of general type can attain arbitrarily large topology regardless of the genus and the number of base points of Lefschetz pencils on them.
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    R. Inanc Baykur, Kenta Hayano
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    ABSTRACT: The purpose of this note is to explain a combinatorial description of closed smooth oriented 4-manifolds in terms of positive Dehn twist factorizations of surface mapping classes, and further explore these connections. This is obtained via monodromy representations of simplified broken Lefschetz fibrations on 4-manifolds, for which we provide an extension of Hurwitz moves that allows us to uniquely determine the isomorphism class of a broken Lefschetz fibration. We furthermore discuss broken Lefschetz fibrations whose monodromies are contained in special subgroups of the mapping class group; namely, the hyperelliptic mapping class group and in the Torelli group, respectively, and present various results on them which extend or contrast with those known to hold for honest Lefschetz fibrations. Lastly, we show that there are 4-manifolds admitting infinitely many pairwise nonisomorphic relatively minimal broken Lefschetz fibrations with isotopic regular fibers.
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    R. Inanc Baykur
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    ABSTRACT: We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who constructed pairs of nonisomorphic Lefschetz fibrations on knot surgered elliptic surfaces. In turn, we prove that there are monodromy factorizations of Lefschetz pencils which have the same characteristic numbers but cannot be obtained from each other via partial conjugations by Dehn twists, answering a problem posed by Auorux.
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    R. Inanc Baykur
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    ABSTRACT: We give a short proof of a conjecture of Stipsicz on the minimality of fiber sums of Lefschetz fibrations, which was proved earlier by Usher. We then construct the first examples of genus g > 1 Lefschetz fibrations on minimal symplectic 4-manifolds which admit unique decompositions as fiber sums.
    Proceedings of the American Mathematical Society 07/2014; DOI:10.1090/proc/12835 · 0.63 Impact Factor
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    R. Inanc Baykur, Kenta Hayano
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    ABSTRACT: The purpose of this article is to initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in mapping class groups of surfaces. We show that, using our methods, one can effectively capture various interesting surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes or the curious 2-section of a genus two Lefschetz fibration which leads to a counter-example to Stipsicz's conjecture on fiber sum indecomposable Lefschetz fibrations.
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    R. Inanc Baykur, Jeremy Van Horn-Morris
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    ABSTRACT: We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book --- except possibly for a few low genera cases. To obtain our results, we produce the first examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary. This solves an open problem posed by Auroux, Smith and Wajnryb, and a more general variant of it raised by Korkmaz, Ozbagci and Stipsicz, independently.
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    R. Inanc Baykur, Dan Margalit
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    ABSTRACT: For each \(g \ge 3\) and \(h \ge 2\), we explicitly construct (1) fiber sum indecomposable relatively minimal genus \(g\) Lefschetz fibrations over genus \(h\) surfaces whose monodromies lie in the Torelli group, (2) genus \(g\) Lefschetz fibrations over genus \(h\) surfaces that are not fiber sums of holomorphic ones, and (3) fiber sum indecomposable genus \(g\) surface bundles over genus \(h\) surfaces whose monodromies are in the Torelli group (provided \(g \ge 4\)). The last result amounts to finding explicit irreducible embeddings of surface groups into Torelli groups; in fact we find such embeddings into arbitrary terms of the Johnson filtration.
    Geometriae Dedicata 10/2012; 177(1). DOI:10.1007/s10711-014-9989-8 · 0.47 Impact Factor
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    R. Inanc Baykur
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    ABSTRACT: We prove that if a closed oriented 4-manifold X fibers over a 2- or 3-dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented 4-manifold X which is not a tower of torus bundles and fibering over a 2- or 3-dimensional manifold does not admit a torsion symplectic canonical class, nor is of Kodaira dimension zero.
    Proceedings of the American Mathematical Society 10/2012; DOI:10.1090/S0002-9939-2014-12151-4 · 0.63 Impact Factor
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    R. Inanc Baykur, Stefan Friedl
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    ABSTRACT: We mostly determine which closed smooth oriented 4-manifolds fibering over smaller non-zero dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds
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    R. Inanc Baykur, Dan Margalit
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    ABSTRACT: For each pair of integers g at least 2 and h at least 1, we explicitly construct infinitely many fiber sum and section sum indecomposable genus g surface bundles over genus h surfaces whose total spaces are pairwise homotopy inequivalent.
    Journal of Topology and Analysis 09/2012; DOI:10.1142/S179352531350009X · 0.34 Impact Factor
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    R. Inanc Baykur, Jeremy Van Horn-Morris
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    ABSTRACT: We show that there are vast families of contact 3-manifolds each member of which admits infinitely many Stein fillings with arbitrarily big euler characteristics and arbitrarily small signatures ---which disproves a conjecture of Stipsicz and Ozbagci. To produce our examples, we set a framework which generalizes the construction of Stein structures on allowable Lefschetz fibrations over the 2-disk to those over any orientable base surface, along with the construction of contact structures via open books on 3-manifolds to spinal open books introduced in [24].
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    R. Inanc Baykur
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    ABSTRACT: The purpose of this article is two-fold: We first give a more elementary proof of a recent theorem of Korkmaz, Monden, and the author, which states that the commutator length of the n-th power of a Dehn twist along a boundary parallel curve on a surface with boundary S of genus g at least two is the floor of (|n|+3)/2 in the mapping class group of S. The alternative proof we provide goes through push maps and Morita's use of Milnor-Wood inequalities, in particular it does not appeal to gauge theory. In turn, we produce infinite families of pairwise non-homotopic 4-manifolds admitting genus g surface bundles over genus h surfaces with distinguished sections which are flat but admit no flat connections for which the sections are flat, for every fixed pairs of integers g and h at least two. The latter result generalizes a theorem of Bestvina, Church, and Souto, and allows us to obtain a simple proof of Morita's non-lifting theorem (for an infinite family of non-conjugate subgroups) in the case of marked surfaces.
    The Michigan Mathematical Journal 06/2012; DOI:10.1307/mmj/1401973053 · 0.65 Impact Factor
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    R. Inanc Baykur
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    ABSTRACT: The broken genera are orientation preserving diffeomorphism invariants of closed oriented 4-manifolds, defined via broken Lefschetz fibrations. We study the properties of the broken genera invariants, and calculate them for various 4-manifolds, while showing that the invariants are sensitive to exotic smooth structures.
    Geometry and Topology Monographs 05/2012; DOI:10.2140/gtm.2012.18.9
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    R. Inanc Baykur
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    ABSTRACT: We show how certain stabilizations produce infinitely many closed oriented 4-manifolds which are the total spaces of genus g surface bundles (resp. Lefschetz fibrations) over genus h surfaces and have non-zero signature, but do not admit complex structures with either orientations, for "most" (resp. all) possible values of g at least 3 and h at least 2 (resp. g at least 2 and h non-negative).
    Mathematical Research Letters 11/2011; DOI:10.4310/MRL.2012.v19.n3.a5 · 0.63 Impact Factor
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    R. Inanc Baykur, Mustafa Korkmaz, Naoyuki Monden
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    ABSTRACT: We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the base genus h are positive, we prove that the adjunction bound 2h-2 is the only universal bound on the self-intersection number of a section of any such genus g bundle and fibration. As a side result, in the mapping class group of a surface with boundary, we calculate the precise value of the commutator lengths of all powers of a Dehn twist about a boundary component, concluding that the stable commutator length of such a Dehn twist is 1/2. We furthermore prove that there is no upper bound on the number of critical points of genus-g Lefschetz fibrations over surfaces with positive genera admitting sections of maximal self-intersection, for g at least two.
    Transactions of the American Mathematical Society 10/2011; DOI:10.1090/S0002-9947-2013-05840-0 · 1.10 Impact Factor
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    R. Inanc Baykur, Masashi Ishida
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    ABSTRACT: In this article, we produce infinite families of 4-manifolds with positive first betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the non-vanishing of the stable cohomotopy Seiberg-Witten invariants of their connected sums. Elementary building blocks used in the earlier work of Ishida and Sasahira are shown to be included in our general construction scheme as well. We then use these families to construct the first examples of families of closed smooth 4-manifolds for which Gromov's simplicial volume is nontrivial, Perelman's lambda-bar invariant is negative, and the relevant Gromov-Hitchin-Thorpe type inequality is satisfied, yet no non-singular solution to the normalized Ricci flow for any initial metric can be obtained. Fang, Zhang and Zhang conjectured that the existence of any non-singular solution to the normalized Ricci flow on smooth 4-manifolds with non-trivial Gromov's simplicial volume and negative Perelman's lambda-bar invariant implies the Gromov-Hitchin-Thorpe type inequality. Our results in particular imply that the converse of this fails to be true for vast families of 4-manifolds. Comment: 46 pages, 1 figure
    Journal of Geometric Analysis 11/2010; 24(4). DOI:10.1007/s12220-013-9392-y · 0.87 Impact Factor
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    R. Inanc Baykur, Seiichi Kamada
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    ABSTRACT: In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identify the 4-manifolds admitting genus one simplified broken Lefschetz fibrations. Comment: 19 pages, 12 figures
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    R. Inanc Baykur
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    ABSTRACT: This note presents the handlebody argument for modifying achiral Lefschetz singularities into broken Lefschetz fibrations, yielding a handlebody proof of the existence of broken Lefschetz fibrations on arbitrary closed smooth oriented 4-manifolds based on the earlier work of Gay and Kirby. Appeared in Geometry and Topology 13 (2009), 312-317; the references are updated herein. Comment: Better to view some figures in color on screen