Publications (18)5.2 Total impact
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ABSTRACT: We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3manifold 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 nonseparating 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.12/2012;  [Show abstract] [Hide abstract]
ABSTRACT: 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.10/2012;  [Show abstract] [Hide abstract]
ABSTRACT: We prove that if a closed oriented 4manifold X fibers over a 2 or 3dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented 4manifold X which is not a tower of torus bundles and fibering over a 2 or 3dimensional manifold does not admit a torsion symplectic canonical class, nor is of Kodaira dimension zero.10/2012;  [Show abstract] [Hide abstract]
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; · 0.67 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We show that there are vast families of contact 3manifolds 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 2disk to those over any orientable base surface, along with the construction of contact structures via open books on 3manifolds to spinal open books introduced in [24].08/2012; 
Article: Flat bundles and commutator lengths
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ABSTRACT: The purpose of this article is twofold: 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 nth 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 MilnorWood inequalities, in particular it does not appeal to gauge theory. In turn, we produce infinite families of pairwise nonhomotopic 4manifolds 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 nonlifting theorem (for an infinite family of nonconjugate subgroups) in the case of marked surfaces.06/2012;  [Show abstract] [Hide abstract]
ABSTRACT: The broken genera are orientation preserving diffeomorphism invariants of closed oriented 4manifolds, defined via broken Lefschetz fibrations. We study the properties of the broken genera invariants, and calculate them for various 4manifolds, while showing that the invariants are sensitive to exotic smooth structures.05/2012;  [Show abstract] [Hide abstract]
ABSTRACT: We show how certain stabilizations produce infinitely many closed oriented 4manifolds which are the total spaces of genus g surface bundles (resp. Lefschetz fibrations) over genus h surfaces and have nonzero 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 nonnegative).11/2011;  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the possible selfintersection 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 2h2 is the only universal bound on the selfintersection 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 genusg Lefschetz fibrations over surfaces with positive genera admitting sections of maximal selfintersection, for g at least two.10/2011;  [Show abstract] [Hide abstract]
ABSTRACT: In this article, we produce infinite families of 4manifolds with positive first betti numbers and meeting certain conditions on their homotopy and smooth types so as to conclude the nonvanishing of the stable cohomotopy SeibergWitten 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 4manifolds for which Gromov's simplicial volume is nontrivial, Perelman's lambdabar invariant is negative, and the relevant GromovHitchinThorpe type inequality is satisfied, yet no nonsingular solution to the normalized Ricci flow for any initial metric can be obtained. Fang, Zhang and Zhang conjectured that the existence of any nonsingular solution to the normalized Ricci flow on smooth 4manifolds with nontrivial Gromov's simplicial volume and negative Perelman's lambdabar invariant implies the GromovHitchinThorpe type inequality. Our results in particular imply that the converse of this fails to be true for vast families of 4manifolds. Comment: 46 pages, 1 figureJournal of Geometric Analysis 11/2010; · 0.86 Impact Factor  [Show abstract] [Hide abstract]
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 2disk with certain nontrivial global monodromies using chart descriptions, and identify the 4manifolds admitting genus one simplified broken Lefschetz fibrations. Comment: 19 pages, 12 figures10/2010;  [Show abstract] [Hide abstract]
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 4manifolds based on the earlier work of Gay and Kirby. Appeared in Geometry and Topology 13 (2009), 312317; the references are updated herein. Comment: Better to view some figures in color on screen09/2010;  [Show abstract] [Hide abstract]
ABSTRACT: Round handles are affiliated with smooth 4manifolds in two major ways: 5dimensional round handles appear extensively as the building blocks in cobordisms between 4manifolds, whereas 4dimensional 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 4manifolds with the same euler characteristics, and if one of them is simplyconnected, then there is a cobordism between them which is composed of round 2handles 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 4dimensional analogue of the LickorishWallace theorem for 3manifolds: Every closed simplyconnected 4manifold 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, while suggesting more constraints on the cobordisms in consideration. We also use round handles to show that every infinite family of mutually nondiffeomorphic closed smooth oriented simplyconnected 4manifolds 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 nonspin, 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 4manifold can be realized as relevant modifications of a broken Lefschetz fibration on it.09/2010;  [Show abstract] [Hide abstract]
ABSTRACT: For each pair $(e,\sigma)$ of integers satisfying $2e+3\sigma\ge 0$, $\sigma\leq 2$, and $e+\sigma\equiv 0\pmod{4}$, with four exceptions, we construct a minimal, simply connected symplectic 4manifold with Euler characteristic $e$ and signature $\sigma$. We also produce simply connected, minimal symplectic 4manifolds with signature zero (resp. signature 1) with Euler characteristic $4k$ (resp. $4k+1$) for all $k\ge 46$ (resp. $k\ge 49$).Journal of the European Mathematical Society 06/2010; · 1.88 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We prove that every closed oriented smooth 4manifold X admits a broken Lefschetz fibration (aka singular Lefschetz fibration) over the 2sphere. 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. We make use of topological modifications of smooth maps with fold and cusp singularities due to Saeki and Levine, and thus we get alternative proofs of previous existence results. Also shown is the existence of broken Lefschetz pencils with connected fibers on any nearsymplectic 4manifold.International Mathematics Research Notices 02/2008; · 1.12 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: The purpose of this article is twofold. First we outline a general construction scheme for producing simplyconnected minimal symplectic 4manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic 4manifolds homeomorphic but not diffeomorphic to $\CP#(2k+1)\CPb$ for $k = 1,...,4$, or to $3\CP# (2l+3)\CPb$ for $l =1,...,6$. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on $\CP#3\CPb$, $3\CP#5\CPb$ and $3\CP#7\CPb$.Journal of Topology 04/2007; volume 1(2):409428. · 0.67 Impact Factor 
Article: Kahler decomposition of 4manifolds
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ABSTRACT: In this article we show that every closed oriented smooth 4manifold 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 4manifolds.02/2006; 
Article: Kähler decomposition of 4–manifolds
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ABSTRACT: In this article we show that every closed oriented smooth 4manifold 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 4manifolds.Algebraic and Geometric Topology  ALGEBR GEOM TOPOL. 01/2006; 6:12391265.
Publication Stats
75  Citations  
5.20  Total Impact Points  
Top Journals
Institutions

2010

Max Planck Institute for Mathematics
Bonn, North RhineWestphalia, Germany
