Publications (25)10.2 Total impact
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ABSTRACT: In this paper, we study stable equivalence of exotically knotted surfaces in 4manifolds, 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 4manifold, 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 simplyconnected 4manifolds. 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.  [Show abstract] [Hide abstract]
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 wellknown 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 blownup pencils.  [Show abstract] [Hide abstract]
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 4manifolds / Stein fillings of contact 3manifolds 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 4manifolds of general type can attain arbitrarily large topology regardless of the genus and the number of base points of Lefschetz pencils on them.  [Show abstract] [Hide abstract]
ABSTRACT: The purpose of this note is to explain a combinatorial description of closed smooth oriented 4manifolds 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 4manifolds, 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 4manifolds admitting infinitely many pairwise nonisomorphic relatively minimal broken Lefschetz fibrations with isotopic regular fibers.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that any symplectic 4manifold which is not a rational or ruled surface, after sufficiently many blowups, 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.  [Show abstract] [Hide abstract]
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 4manifolds which admit unique decompositions as fiber sums.Proceedings of the American Mathematical Society 07/2014; DOI:10.1090/proc/12835 · 0.63 Impact Factor  [Show abstract] [Hide abstract]
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 4manifolds as multisections, such as SeibergWitten basic classes or the curious 2section of a genus two Lefschetz fibration which leads to a counterexample to Stipsicz's conjecture on fiber sum indecomposable Lefschetz fibrations.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
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/s1071101499898 · 0.47 Impact Factor  [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.Proceedings of the American Mathematical Society 10/2012; DOI:10.1090/S000299392014121514 · 0.63 Impact Factor  [Show abstract] [Hide abstract]
ABSTRACT: We mostly determine which closed smooth oriented 4manifolds fibering over smaller nonzero dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4manifolds  [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; DOI:10.1142/S179352531350009X · 0.34 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]. 
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.The Michigan Mathematical Journal 06/2012; DOI:10.1307/mmj/1401973053 · 0.65 Impact Factor  [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.Geometry and Topology Monographs 05/2012; DOI:10.2140/gtm.2012.18.9  [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).Mathematical Research Letters 11/2011; DOI:10.4310/MRL.2012.v19.n3.a5 · 0.63 Impact Factor  [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.Transactions of the American Mathematical Society 10/2011; DOI:10.1090/S000299472013058400 · 1.10 Impact Factor  [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; 24(4). DOI:10.1007/s122200139392y · 0.87 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 figures  [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 screen
Publication Stats
138  Citations  
10.20  Total Impact Points  
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Institutions

2010–2012

Brandeis University
 Department of Mathematics
Волтам, Massachusetts, United States


2006

Michigan State University
ИстЛансинг, Michigan, United States
