Research

My research interests range over low dimensional geometry and topology, focusing on smooth 4-manifolds. In particular, I am interested in constructions of contact structures on 3-manifolds, and (exotic) smooth, symplectic, and complex structures on 4-manifolds, as well as in descriptions and study of 4-manifolds via various singular fibrations, where Gauge theory and mapping class groups play vital roles.

Below is a list of my research publications, preprints, and books, along with extended abstracts and pdf links, mostly to the respective versions on the arXiv.

Publications

  • Positive factorizations of mapping classes
    (joint with N. Monden and J. Van Horn-Morris)
    Algebraic & Geometric Topology, Vol. 17 (2017), no. 3, 1527-1555.
    In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers g,n, we tell whether or not there exist genus-g Lefschetz pencils with n base points, and if there are, what the maximal Euler characteristic is whenever it is bounded above. 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.
  • Small Lefschetz fibrations and exotic 4-manifolds
    (joint with M. Korkmaz)
    Mathematische Annalen, Vol. 367 (2017), no. 3-4, 1333-1361.
    We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP^2 # p (-CP^2) for p=7, 8, 9, and to 3 CP^2 #q (-CP^2) for q =12,...,19. Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with b^+ at most 3, with one possible exception when b^+=3. Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smallest possible example, which follow from a reverse engineering procedure we introduce for this setting. We also derive exotic minimal symplectic 4-manifolds in the homeomorphism classes of CP^2 # 4 (-CP^2) and 3 CP^2 # 6 (-CP^2) from small Lefschetz fibrations over surfaces of higher genera.
  • Knotted surfaces in 4-manifolds and stabilizations
    (joint with N. Sunukjian)
    Journal of Topology, Vol. 9 (2016), no. 1, 215-231.
    In this paper, we study stable equivalence of exotically knotted surfaces in 4-manifolds, surfaces that are topologically isotopic, but not smoothly. 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 morever 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.
  • Multisections of Lefschetz fibrations and topology of symplectic 4-manifolds
    (joint with K. Hayano)
    Geometry & Topology, Vol. 20 (2016), No. 4, 2335-2395.
    We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes and exceptional classes, or branched loci of compact Stein surfaces as branched coverings of the 4-ball. Various problems regarding the topology of symplectic 4-manifolds, such as the smooth classification of symplectic Calabi-Yau 4-manifolds, can be translated to combinatorial problems in this manner. After producing special monodromy factorizations of Lefschetz pencils on symplectic Calabi-Yau K3 and Enriques surfaces, and introducing monodromy substitutions tailored for generating multisections, we obtain several novel applications, allowing us to construct: new counter-examples to Stipsicz's conjecture on fiber sum indecomposable Lefschetz fibrations, non-isomorphic Lefschetz pencils of the same genera on the same new symplectic 4-manifolds, the very first examples of exotic Lefschetz pencils, and new exotic embeddings of surfaces.
  • Fillings of genus-1 open books and 4-braids
    (joint with J. Van Horn-Morris)
    International Mathematics Research Notices, (2016), doi:https://doi.org/10.1093/imrn/rnw281.
    We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings, but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary contact 3-manifold. In contrast, we observe that there are only finitely many possibilities for the homology groups of Stein fillings of a given contact 3-manifold with support genus zero. We also show that there are 4-strand braids which admit infinitely many distinct Hurwitz classes of quasipositive factorizations, yielding in particular an infinite family of knotted complex analytic annuli in the 4-ball bounding the same transverse link up to transverse isotopy. These realize the smallest possible examples in terms of the number of boundary components a genus-1 mapping class and the number of strands a braid can have with infinitely many positive/quasipositive factorizations.
  • Inequivalent Lefschetz fibrations and surgery equivalence of symplectic 4-manifolds
    Journal of Symplectic Geometry, Vol. 14 (2016), No. 3, 671-686.
    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.
  • Topological complexity of symplectic 4-manifolds and Stein fillings
    (joint with J. Van Horn-Morris)
    Journal of Symplectic Geometry, Vol. 14 (2016), no. 1, 171-202.
    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.
  • Minimality and fiber sum decompositions of Lefschetz fibrations
    Proceedings of the American Mathematical Society, 144 (2016), no. 5, 2275-2284.
    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, up to diffeomorphisms of the summands, admit unique decompositions as fiber sums.
  • Hurwitz equivalence for Lefschetz fibrations and their multisections
    (joint with K. Hayano)
    Contemporary Mathematics, 675 (2016), 1-24, Amer. Math. Soc., Providence, RI.
    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. This article supplements our work in [1].
  • Families of contact 3-manifolds with arbitrarily large Stein fillings
    (joint with J. Van Horn-Morris), with an appendix by S. Lisi and C. Wendl
    Journal of Differential Geometry, Vol. 101 (2015), No. 3, 423-465.
    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. Lisi and Wendl wrote a very nice appendix to our article, which presents their proof of the existence of Stein structures on allowable Lefschetz fibrations over general surfaces using a variation of Gompf-Thurston construction.
  • Broken Lefschetz fibrations and mapping class groups
    (joint with K. Hayano)
    Geometry & Topology Monographs, Vol. 19 (2015) 269–290.
    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, in 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 observe that there are infinitely many pairwise nonisomorphic broken Lefschetz fibrations with smoothly isotopic regular fibers.
  • Virtually symplectic fibered 4-manifolds
    (joint with S. Friedl)
    Indiana University Mathematics Journal, Vol. 64 (2015), Issue 4, 983-999.
    We mostly determine which closed oriented 4-manifolds fibering over smaller non-zero dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds. The case of mapping tori of 3-manifolds, which is the most challenging when the 3-manifold is reducible, is further analyzed in terms of their virtual) symplectic Kodaira dimension by Li and Ni in [1].
  • Lefschetz fibrations and Torelli groups
    (joint with D. Margalit)
    Geometriae Dedicata, Vol. 177 (2015), No. 1, 275-291.
    For each g > 2 and h > 1, 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) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided g > 3), and (3) infinitely many genus g Lefschetz fibrations over genus h surfaces that are not fiber sums of holomorphic ones. A variation of our argument (observed by Korkmaz) is recently employed by Salter to generate examples of 4-manifolds admitting distinct surface bundles over surfaces [1] .
  • Classification of broken Lefschetz fibrations with small fiber genera
    (joint with S. Kamada)
    Journal of the Mathematical Society of Japan, Vol. 67 (2015), No. 3, 877-901.
    In this article, we generalized the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classified genus one Lefschetz fibrations over the 2-disk with certain non-trivial global monodromies using chart descriptions, and identified the 4-manifolds admitting genus one simplified broken Lefschetz fibrations up to blow-ups. Since then, several closely related partial classification results with additional constraints (yet without appealing to blow-ups) appeared in the literature: (i) in [1] Hayano independently classified genus one simplified BLFs with at most six Lefschetz critical points; (ii) in [2] he classified genus one simplified BLFs on simply-connected spin 4-manifolds, and (iii) in [3] Behrens classified indefinite generic maps with cusps that can be transformed by a ''face-lift'' into a proper subclass of genus one simplified BLFs. The classification result obtained in our article was also used to estimate and calculate broken genera invariants of various smooth 4-manifolds in [4] .
  • Virtual betti numbers and the symplectic Kodaira dimension of fibered 4-manifolds
    Proceedings of the American Mathematical Society, Vol. 142 (2014), 4377-4384.
    We prove that if a closed smooth 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. Subsequently, Tian-Jun Li and Yi Ni obtained an analogous result for virtual Betti numbers of 4-manifolds fibering over the circle with irreducible fibers [1]. When combined together, there results now manifest that virtual Betti number of (almost all) fibered 4-manifolds behave the same way as those of smaller dimensional manifolds.
  • Families of 4-manifolds with nontrivial Seiberg-Witten stable cohomotopy invariants and normalized Ricci flow
    (joint with M. Ishida)
    Journal of Geometric Analysis, Vol. 24 (2014), No. 4, 1716-1736.
    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.
  • Flat bundles and commutator lengths
    Michigan Mathematical Journal, Vol. 63 (2014), No. 2, 333-344.
    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 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.
  • Sections of surface bundles and Lefschetz fibrations
    (joint with M. Korkmaz and N. Monden),
    Transactions of the American Mathematical Society, Vol. 365 (2013), No. 11, 5999-6016.
    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. The mapping class group relations we have discovered in this article led to a new proof of Morita's non-lifting theorem (from MCG to Diff) for many subgroups [1], as well as to constructions of families of contact 3-manifolds with arbitrarily large Stein fillings (disproving a couple of popular conjectures by Stipsicz and Ozbagci) by Jeremy Van-Horn Morris and myself [2].
  • Indecomposable surface bundles over surfaces
    (joint with D. Margalit)
    Journal of Topology and Analysis, Vol. 3 (2013), No. 2, 161-181.
    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. This article settles the question on the existence of indecomposable surface bundles in the most satisfactory way: for all possible genera (since the answer is known to be negative for smaller fiber and base genera) and for infinite families of 4-manifolds, by providing algebraic descriptions for fiber sum and section sum indecomposability, and then relying on the advances in geometric group theory on embeddings of surface groups into mapping class groups.
  • Round handles, logarithmic transforms and smooth 4-manifolds
    (joint with N. Sunukjian),
    Journal of Topology, Vol. 6 (2013), No. 1, 49-63.
    Round handles are affiliated with smooth 4-manifolds in two major ways: 5-dimensional round handles appear extensively as the building blocks in cobordisms between 4-manifolds, whereas 4-dimensional round handles are the building blocks of broken Lefschetz fibrations on them. The purpose of this article is to shed more light on these interactions. We prove that if X and X' are cobordant closed smooth 4-manifolds with the same euler characteristics, and if one of them is simply-connected, then there is a cobordism between them which is composed of round 2-handles only, and therefore one can pass from one to the other via a sequence of generalized logarithmic transforms along tori. As a corollary, we obtain a new proof of a theorem of Iwase's, which is a 4-dimensional analogue of the Lickorish-Wallace theorem for 3-manifolds: Every closed simply-connected 4-manifold can be produced by a surgery along a disjoint union of tori contained in a connected sum of copies of CP^2, -CP^2 and S^1 x S^3. These answer some of the open problems posted by Ron Stern [1], while suggesting more constraints on the cobordisms in consideration. We also use round handles to show that every infinite family of mutually non-diffeomorphic closed smooth oriented simply-connected 4-manifolds in the same homeomorphism class constructed up to date consists of members that become diffeomorphic after one stabilization with S^2 x S^2 if members are all non-spin, and with S^2 x S^2 # -CP^2 if they are spin. In particular, we show that simple cobordisms exist between knot surgered manifolds. We then show that generalized logarithmic transforms can be seen as standard logarithmic transforms along fiber components of broken Lefschetz fibrations, and show how changing the smooth structures on a fixed homeomorphism class of a closed smooth 4-manifold can be realized as relevant modifications of a broken Lefschetz fibration on it. This is the article with the longest abstract I have ever written!
  • Non-holomorphic surface bundles and Lefschetz fibrations
    Mathematical Research Letters, Vol. 19 (2012), No. 3, 567-574.
    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). This article answers a more extensive version of the MathOverflow question of Jim Bryan's, who asked whether every surface bundle could be made holomorphic, provided the signature of the total space was non-zero.
  • Broken Lefschetz fibrations and smooth structures on 4-manifolds
    Geometry & Topology Monographs, Vol. 12 (2012), 9-34.
    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. The purpose is two-fold: we aim to initiate a study of smooth structures on 4-manifolds (and smooth embeddings of surfaces) via broken genus invariants, and meanwhile suggest a systematic framework for the rapidly growing literature on the topology of broken Lefschetz fibrations on 4-manifolds by considering simplified broken Lefschetz fibrations of fixed genera.
  • Simply connected minimal symplectic 4-manifolds with signature less than -1
    (joint with A. Akhmedov, S. Baldridge, P. Kirk and B. D. Park),
    Journal of the European Mathematical Society, Vol. 12 (2010), No. 1, 133-161.
    The symplectic geography problem asks which pairs of integers can be realized as the euler characteristics and signatures of minimal symplectic 4-manifolds with fixed fundamental group, typically the trivial group. This article presents a systematic way to populate the largest portion of the geography plane up to date, extending the pioneering works of Gompf, J. Park, Fintushel-Stern, Stipsicz, and others. Namely, for each pair (e,s) of integers such that 2e+3s non-negative, e+s divisible by four, and s < -1, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic e and signature s. (The first two conditions hold a priori for any such 4-manifold, whereas the last one is a posteriori - admittedly marking the limitations of our knowledge and skills on how to construct symplectic 4-manifolds with non-negative signatures.) We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature -1) with Euler characteristic 4k (resp. 4k+1) for k smaller than 50. The families of minimal symplectic 4-manifolds we have constructed in this article, at times along with the techniques we developed, were used in succeeding works of Akhmedov and D. Park (who managed to fill a few missing lattice points not mentioned above), Torres, Usher, and Yazinski, as well as M. Ishida and myself.
  • Topology of broken Lefschetz fibrations and near-symplectic 4-manifolds
    Pacific Journal of Mathematics, Vol. 240 (2009), No. 2, 201-230.
    Did you know that any closed smooth oriented 4-manifold could be described by an ordered collection of simple closed curves on a closed orientable surface? This article provides such a description via so-called simplified broken Lefschetz fibrations on 4-manifolds with b^+>0, which shortly after was seen to exist on all closed oriented smooth 4-manifolds (the papers below actually appeared later, but were published faster). In this article, the topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles, and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefschetz fibrations/pencils, called "simplified BLF/BLPs", while showing that all near-symplectic closed 4-manifolds can be supported by these a la Auroux, Donaldson, Katzarkov. Various constructions of broken Lefschetz fibrations and a generalization of the symplectic fiber sum operation to the near-symplectic setting are given. Extending the study of Lefschetz fibrations, we detect certain constraints on the symplectic fiber sum operation to result in a 4-manifold with nontrivial Seiberg-Witten invariant, as well as the self-intersection numbers that sections of broken Lefschetz fibrations can acquire. The description of a near-symplectic 4-manifold (equivalently a 4-manifold with b^+>0) in terms of a product of positive Dehn twists about simple curves and a distinguished curve on an orientable surface, along with a special handlebody diagram, which allows one to effectively identify the underlying 4-manifold, was employed successfully in different works by Hayano, M. Sato, Kamada and myself in order to address several problems surrounding BLFs. The very same description, with a simple translation to indefinite generic maps, led to the study of 4-manifolds via surface diagrams by Behrens, Hayano, and J. Williams, following the subsequent work on the existence of BLFs on arbitrary 4-manifolds by myself and others.
  • Handlebody argument for modifying achiral singularities
    Geometry & Topology, Vol. 13 (2009), 312-317.
    This note details a handlebody argument for modifying achiral Lefschetz singularities into broken Lefschetz fibrations, which first appeared in my IMRN paper on BLFs (See Example 3.5 in [1]), 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. A different handlebody proof, which overrides the Gay-Kirby construction of achiral BLFS that inspired many of us, was given later by Akbulut and Karakurt. This note appeared as an appendix to Lekili's "Wrinkled fibrations on smooth manifolds".
  • Existence of broken Lefschetz fibrations
    International Mathematics Research Notices (2008), Art. ID rnn 101, 15 pp.
    We prove that every closed oriented smooth 4-manifold X admits a broken Lefschetz fibration (aka singular Lefschetz fibration) over the 2-sphere. Given any closed orientable surface F of square zero in X, we can choose the fibration so that F is a fiber. Moreover, we can arrange it so that there is only one Lefschetz critical point when the Euler characteristic e(X) is odd, and none when e(X) is even. This article provided the first proof of the existence of broken Lefschetz fibrations on arbitrary closed oriented smooth 4-manifolds, disproving a conjecture of Gay and Kirby on the necessity of negative Lefschetz critical points to achieve this general existence result. It moreover presented a very short and simple proof of a theorem of Auroux, Donaldson, Katzarkov, who proved the existence of broken Lefschetz pencils on 4-manifolds with b^+>0 using approximately holomorphic theory. Both of my proofs were given via elementary topological modifications of the singularities of generic maps, following the beautiful works of Thom, Levine and Saeki. Shortly after the appearance of my paper [1], the existence of BLFs on arbitrary 4-manifolds, using the Gay-Kirby construction of achiral BLFs, was also shown by Y. Lekili in an update of his arxiv paper [2], and a couple months later by Akbulut and Karakurt using different methods [3]. The flip and slip move I introduced in the paper so as to prove the existence of BLFs and BLPs with connected fibers was later studied by Williams [4] and Hayano [5] as a key "stabilization" move for simplified BLFs and simplified indefinite generic maps.
  • Constructing infinitely many smooth structures on small 4-manifolds
    (joint with A. Akhmedov and B. D. Park),
    Journal of Topology, Vol. 2 (2008), 409-428.
    Constructing exotic smooth structures on simply-connected 4-manifolds with small euler characteristics is known to be most challenging, due to the shortage of interesting embedded surfaces in them. Here we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics, which is a very much simplified instance of Fintushel-Stern's reverse engineering procedure [1] . Using this scheme, we illustrate how to obtain irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to CP^2#(2k+1)(-CP^2) for k = 1,...,4, or to 3CP^2# (2l+3)(-CP^2) for l =1,...,6. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on CP^2#3(-CP^2), 3 CP^2#5 (-CP^2) and 3 CP^2#7(-CP^2). This article featured a key observation from my thesis which related seemingly different constructions of the exotic CP^2 # 3 (-CP^2)s by Akhmedov and Park in [2] and Baldridge and Kirk in [3], which in turn allowed us to vary the construction to obtain infinite families of exotic smooth structures. The construction scheme we developed here was later employed by Akhmedov and Park to construct similar families on even blow-ups of CP^2 and 3 CP^2 in [4] using the building block for the exotic CP^2 # (-2 CP^2) they came up with. The manifolds we have constructed were later used in a nice paper of Rasdeaconu and Suvaina [5] to exhibit the existence of smooth structures on blow-ups of CP^2 which do not support Einstein metrics, even though there are pairs of smooth structures on the same topological manifolds which admit Einstein metrics of positive and negative scalar curvatures, respectively.
  • Kähler decomposition of 4-manifolds
    Algebraic & Geometric Topology, Vol. 6 (2006), 1239-1265.
    This is my first research article which I wrote while I was a graduate student at Michigan State. Here we show that every closed oriented smooth 4-manifold can be decomposed into two codimension zero submanifolds (one with reversed orientation) so that both pieces are exact Kahler manifolds with strictly pseudoconvex boundaries and that induced contact structures on the common boundary are isotopic. Meanwhile, matching pairs of Lefschetz fibrations with bounded fibers are offered as the geometric counterpart of these structures, analogous to Donaldson's Lefschetz pencils on symplectic 4-manifolds. We also provide a simple topological proof of the existence of folded symplectic forms on 4-manifolds, a result originally due to Cannas da Silva [1]. The extra features I have got allow one to get so-called folded-Kahler forms on arbitrary 4-manifolds, which shows that the curious work of von Bergmann's on the pseudo-holomorphic maps into arbitrary 4-manifolds [2] is not null, so to speak. These folded-Kahler forms were used by Geiges and Stipsicz in [3] who gave a sleek proof of the existence of contact structures on 5-manifolds which are products of S^1 with 4-manifolds, which in turn, led to the proof of the existence of contact structures on all principal S^1 bundles over 4-manifolds by Ding and Geiges in [4].
  • Symplectic structures, Lefschetz fibrations, and their generalizations on smooth 4-manifolds
    Ph.D.Thesis in Mathematics, Michigan State University, East Lansing, MI, June 2007.
    Advisor: Professor Ronald Fintushel
    Thesis abstract: In this thesis, we study symplectic structures, Lefschetz fibrations, and their various generalizations on smooth 4-manifolds along with the associated (smooth) invariants. Our results will be presented in separate chapters as follows: In Chapter 2, we outline a general construction scheme to obtain minimal symplectic structures on simply-connected 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain minimal symplectic 4-manifolds homeomorphic to CP^2 #(2k + 1) (-CP^2) for k = 1,..., 4, or to 3 CP^2 #(2l + 3) (-CP^2) , for l = 2..., 6. Secondly, for each of these homeomorphism types with b+ = 1; we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on CP2#3(-CP^2). In Chapter 3, we study the 4-manifolds with nontrivial Seiberg-Witten invariants which are equipped with near-symplectic broken Lefschetz fibrations. We first study the topology of these fibrations and describe simple presentations of them. We then provide several examples using handlebody diagrams. We define a near-symplectic operation that generalizes the symplectic fiber sum operation, together with its effect on the Seiberg-Witten invariants and Perutz's Lagrangian matching invariants. These techniques are then used to obtain several results on near-symplectic manifolds with non-trivial invariants In Chapter 4, we show that every closed oriented smooth 4-manifold can be decomposed into two codimension zero submanifolds (one with reversed orientation) so that both pieces are exact Kahler manifolds with strictly pseudoconvex boundaries and that induced contact structures on the common boundary are isotopic. Meanwhile, matching pairs of Lefschetz fibrations with bounded fibers are offered as the geometric counterpart of these structures. We also provide a simple topological proof of the existence of folded symplectic forms on 4-manifolds. Finally in the Addendum, we provide answers to two open questions stated by David Gay and Rob Kirby.
  • Approximating smooth maps by real algebraic morphisms
    M.Sc.Thesis in Mathematics, Middle East Technical University, Ankara, Turkey, June 2002.
    Advisor: Professor Yildiray Ozan
    This thesis is from the time I was much interested in real algebraic geometry. Thesis abstract: Given two nonsingular real algebraic varieties, we can consider them as smooth manifolds and view regular maps between them as a subset of the topological space of smooth mappings between them. Thus we can ask when can a smooth map be approximated by algebraic ones. In this thesis, we deal with sufficient and necessary conditions for the set of regular maps to be dense in the smooth mappings, based on two main results of J. Bochnak and W. Kucharz.

Preprints

    This section is not up to date. If you are looking for a newer version of the preprints below, or any preprint in preparation I mentioned in a talk or a paper, best is to e-mail me.

  • Simplified broken Lefschetz fibrations and trisections of 4-manifolds
    (joint with O. Saeki)
    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. 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. This article complements our more extensive work in [1] with further exposition and constructions of small genera simplified trisections, along with alternate proofs of the correspondence between simplified broken Lefschetz fibrations and trisections of 4-manifolds.
  • Simplifying indefinite fibrations on 4-manifolds
    (joint with O. Saeki)
    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.
  • Dissolving knot surgered 4-manifolds by classical cobordism arguments
    The purpose of this short 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.
  • Small symplectic Calabi-Yau surfaces and exotic 4-manifolds via genus-3 pencils
    We explicitly produce symplectic genus-3 Lefschetz pencils (with base points), whose total spaces are homeomorphic but not diffeomorphic to rational surfaces CP^2 # p (-CP^2) for p= 7, 8, 9. We then give a new construction of an infinite family of symplectic Calabi-Yau surfaces with first Betti number b_1=2,3, along with a surface with b_1=4 homeomorphic to the 4-torus. These are presented as the total spaces of symplectic genus-3 Lefschetz pencils we construct via new positive factorizations in the mapping class group of a genus-3 surface. Our techniques in addition allow us to answer in the negative a question of Korkmaz regarding the upper bound on b_1 of a genus-g fibration. (Under revision. Some stronger results are to be incorporated.)
  • An interesting genus-3 Lefschetz fibration
    (joint with M. Korkmaz)
    In this note, we describe a new hyperelliptic genus-3 Lefschetz fibration, which is the smallest of its kind. Our construction yields several interesting questions. (Under revision. This note is now part of a larger article project.)

Books

Interactions between low dimensional topology and mapping class groups "Interactions between low dimensional topology and mapping class groups"

(Proceedings of the Bonn Conference, July 2013)

R. Inanc Baykur, John Etnyre and Ursula Hamenstädt (Editors)
Geometry & Topology Monographs, Volume 19 (2015).

There has been a long history of rich and subtle connections between low dimensional topology, mapping class groups and geometric group theory. From July 1 to July 5, 2013, the conference “Interactions between low dimensional topology and mapping class groups” held at the Max Planck Institute for Mathematics in Bonn highlighted these diverse connections, and fostered new and unexpected collaborations between researchers in these areas. The proceedings for this conference aims to further draw attention to the beautiful mathematics emerging from diverse interactions between these areas. The articles collected in this volume, in addition to gathering new results, also contain expositions and surveys of the latest developments in various active areas of research at the interface of mapping class groups of surfaces and the topology and geometry of 3– and 4–dimensional manifolds. Many open problems and new directions for research are discussed.

Damaya Güzelleme Cover "Celebration of Draughts"

("Damaya Güzelleme", in Turkish)

R. Inanc Baykur, Yazı-Görüntü-Ses Publications, Istanbul, Turkey, November 2007.

This is a book on traditional draughts game, aka "dama", a unique version of checkers considered as the closest cousin of chess. Contains anthropological analysis, visual recordings, short stories, and a detailed treatment of tactics and strategies in the game.

It was featured in the 2016 Agenda "Games" by the Hrant Dink Foundation. Here is an English excerpt from the agenda: "[Dama's] history stretches as far back as the Ur, Egyptian, and Ancient Greek civilizations. It is a unique and innovative mind game known for the multiple possibilities of moves and a dynamism which engenders ample opportunity for the effective use of imagination. There are approximately ten simple rules for moves to be organized collectively with pieces which by themselves have limited power. Sacrifices are the main means for gains. Renowned master players from different backgrounds like shoeshine man Ibrahim Bey, the rival of Sultan Abdulaziz, became famous more for the refinement of their games than their invincibility."