David Gay, Thomas E. Mark
We show that under appropriate hypotheses, a plumbing of symplectic surfaces in a symplectic 4-manifold admits strongly convex neighborhoods. Moreover the neighborhoods are Lefschetz fibered with an easily-described open book on the boundary supporting the induced contact structure. We point out some applications to cut-and-paste constructions of symplectic 4-manifolds.
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Lefschetz fibrations on cotangent bundles and some plumbings
本文提出从Weinstein流形的Weinstein handle分解构造Lefschetz fibration的思想，证明了其在余切丛与某些plumbing情形的定理，并得出$A_{4k+3}$与$D_{4k+3}$奇点Milnor fiber在复维数为奇数时微分同胚的推论。
Fiber sums of genus 2 Lefschetz fibrations
利用Siebert和Tian最近关于亏格为2的Lefschetz纤维化的不可约奇异纤维的全纯性的结果，我们证明了任何亏格为2的Lefschetz纤维化在与全纯纤维化纤维和之后都是全纯的.
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.
We give a maximal set of disjoint $(-1)$-sections of the well-known Lefschetz fibration constructed by Matsumoto, Cadavid and Korkmaz. In fact, we obtain several such sets for a fixed genus, which implies that the Matsumoto-Cadavid-Korkmaz Lefschetz fibration has more than one supporting minimal Lefschetz pencils. We also determine the diffeomorphism types of the obtained supporting minimal Lefschetz pencils.
It was shown by Usher that any fiber sum of Lefschetz fibrations over $S^2$ is minimal, which was conjectured by Stipsicz. We prove that the converse does not hold by showing that there exists an indecomposable minimal genus-2 Lefschetz fibration.
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.
Realizing 4-manifolds as achiral Lefschetz fibrations
我们表明，任何4流形，手术后的曲线，承认非手性莱夫谢茨纤维化。特别地，我们证明了在2-球面上具有2-球面丛的任何单连通4-流形的连通和将允许非手性Lefschetz纤维化。我们还表明，这些surgered流形承认近辛结构，并证明更一般的非手性Lefschetz纤维化的部分有近辛结构。作为一个推论，我们的证明，我们得到一个替代证明Gompf的结果存在的辛结构的Lefschetz铅笔。
We describe Lefschetz-Bott fibrations on complex line bundles over symplectic manifolds explicitly. As an application, we construct more than one strong symplectic filling of the link of the $A_{k}$-type singularity. In the appendix, we show that the total space of a Lefschetz-Bott fibration over the unit disk serves as a strong symplectic filling of a contact manifold compatible with an open book induced by the fibration.
Questions of geography of various classes of $4$-manifolds have been a central motivating question in $4$-manifold topology. Baykur and Korkmaz asked which small, simply connected, minimal $4$-manifolds admit a genus $2$ Lefschetz fibration. They were able to classify all the possible homeomorphism types and realize all but one with the exception of a genus $2$ Lefschetz fibration on a symplectic $4$-manifold homeomorphic, but not diffeomorphic to $3 \mathbb{CP}^2 \# 11\overline{\mathbb{CP}}^2$. We give a positive factorization of type $(10,10)$ that corresponds to such a genus $2$ Lefschetz f…
Topology of broken Lefschetz fibrations and near-symplectic 4-manifolds
利用柄分解研究了断裂Lefschetz纤维化的拓扑结构。我们考虑一个轻微的概括的圆形手柄，并描述所有出现在四维的手柄图。我们建立了简化的体和monodromy表示的某个子类的断裂Lefschetz纤维/铅笔，同时表明，所有的近辛闭4-流形可以支持这些一个拉Auroux，唐纳森，Katzarkov。给出了断裂Lefschetz纤维化的各种构造以及辛纤维和运算在近辛情形下的推广。扩展Lefschetz纤维化的研究，我们发现辛纤维和操作的某些限制，导致一个4-流形与非平凡的Seiberg-Witten不变量，以及自相交数，部分断裂的Lefschetz纤维化可以获得。
Trisections and Lefschetz fibrations with $(-n)$-sections
本文研究了带有(-n)-截面的非定向Lefschetz纤维化的三重分割构造方法，并给出了相关三重分割图的显式构造。
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.
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; n…
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 …
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.
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.
Spin Lefschetz fibrations are abundant
该研究证明有限表现群可作为2-球面上spin Lefschetz纤维化的基本群，且辛地理平面Noether线下方的容许格点可由单连通spin Lefschetz纤维化实现，拓展了相关拓扑结构的存在性结论。
On the Slope of Hyperelliptic Lefschetz Fibrations and the Number of Separating Vanishing Cycles
在这篇文章中，我们找到了一个上，下界的斜率的亏格g超椭圆Lefschetz纤维化，这是尖锐的g = 2时，并证明了强连接，在一般情况下，超椭圆亏格g Lefschetz纤维化的斜率和分离消失圈的数量。具体地说，我们表明，斜率是大于4-4/g当且仅当纤维化包含分离消失的循环。对于亏格g>1的超椭圆Lefschetz纤维化，我们还改进了已有的s/n（分离消失环数与非分离消失环数之比）的界.特别地，我们证明了当g>5时，对于这样的纤维化，s<=n。
Noncomplex smooth 4-manifolds with Lefschetz fibrations
对于任何大于或等于2的整数g，存在无穷多个两两非同胚光滑4-流形，它们允许S^2上的亏格g Lefschetz纤维化，但不携带任何具有任一方向的复结构。这推广了Ozbagci和Stipsicz最近的一个结果。
Sections, pseudosections, and multisections of Lefschetz fibrations
本文解决了辛 Lefschetz 纤维化是否总有光滑截面的长期问题，通过点推映射提升方法证明并非如此，但所有此类纤维化都有正多截面，并构造了奇数亏格≥7的签名零纤维化。
We construct Lefschetz fibrations over $S^2$ which do not satisfy the slope inequality. This gives a negative answer to a question of Hain.
We prove that a positive allowable Lefschetz fibration, PALF in short, admits a structure of exact Lefschetz fibration in the sense of Seidel \cite{Se08}. If the two-fold first Chern class of the total space is zero, we obtain the Fukaya-Seidel category. We prove that the derived Fukaya-Seidel category of PALF is independent of the choice of the symplectic structure. At the end of this paper, we study examples and show that derived Fukaya-Seidel categories have more information than the Milnor lattices of PALFs.
Constructing Lefschetz-type fibrations on four-manifolds
我们展示了如何在任意光滑的、封闭的、定向的4-流形上构造断裂的非手性Lefschetz纤维化。这些是2-球面上的Lefschetz纤维化的推广，其中我们允许具有非标准取向的Lefschetz奇点以及对应于圆形1-手柄的奇点圆。我们也可以安排一个给定的面积为0的正方形是一个纤维。在没有3-和4-柄的4-流形的二重流形的情况下，这种构造更容易和更明确，例如由平凡群的非平凡平衡表示产生的同伦4-球面。
Compatible Relative Lefschetz Fibrations On Admissible Relative Stein Pairs
本文将四维球面Lefschetz纤维化相关结果推广至六维情形，构造了以(p,q,2)-Brieskorn簇为页的六维球面相容Lefschetz纤维化，并在具可容许相对Stein对结构的六维Stein域上构建相对Lefschetz纤维化，给出纯拓扑存在性证明。
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 twi…
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 smal…
Nadler introduced certain Lagrangian singularities indexed by trees, and determined their microlocal sheaves to be the category of modules over the corresponding tree quiver. Another family of spaces indexed by trees: the tree plumbings of spheres. The Fukaya-Seidel category of the Lefschetz fibration with this plumbing as fiber and all spheres as vanishing cycles is well known to also be modules over the tree quiver. Here we upgrade this matching of categories to a matching of geometry.
Existence of broken Lefschetz fibrations
我们证明了每一个闭的光滑4-流形X在2-球面上都有一个破的Lefschetz纤维化（也称为奇异Lefschetz纤维化）。给定X中任意平方为零的闭可定向曲面F，我们可以选择纤维化使得F是纤维。此外，我们可以安排它，使得当欧拉示性数e（X）为奇数时，只有一个莱夫谢茨临界点，而当e（X）为偶数时，没有莱夫谢茨临界点。我们利用拓扑修改的光滑映射的折叠和尖点奇异性，由于佐伯和莱文，从而我们得到以前的存在性结果的替代证明。还表明是存在的断裂Lefschetz铅笔与连接纤维上的任何近辛4-流形。
Lefschetz fibrations with infinitely many sections
本文证明了与全纯情形不同，光滑或辛 Lefschetz 纤维化可具有无穷多个截面，给出了辛情形下存在无穷多同调不同截面的一般判据与实例，并指出在自同构群作用下计数也无法恢复有限性。
In this article, we generalize the results discussed in [arXiv:1004.3762] by introducing a genus to generic fibers of Lefschetz fibrations. That is, we give families of relations in the mapping class groups of genus-1 surfaces with boundaries that represent rational homology disk smoothings of weighted homogeneous surface singularities whose resolution graphs are $3$-legged with a bad central vertex.
Commutators, Lefschetz fibrations and the signatures of surface bundles
我们构造的Lefschetz纤维与规定的奇异纤维的例子。通过采取对这样的纤维化与相同的奇异纤维的差异，我们得到了新的例子，表面丛的表面与非零签名。从这些，我们得到新的上界的最小曲面的亏格表示一个给定的元素在第二个同源的映射类群。
Hyperelliptic Lefschetz fibrations and branched covering spaces
设M是光滑的4-流形，且在2-球面上存在相对极小的超椭圆亏格h-Lefschetz纤维化.如果这个纤维化的所有消失圈都是非分离曲线，那么我们证明了M是2-球面上的2-球面丛的2-重覆盖，在嵌入曲面上分支。如果这个纤维化的消失周期的集合包括分离曲线，我们表明，M是一个Lefshetz纤维化N的相对最小化，与N的2倍分支覆盖的理性表面，分支的嵌入式表面。
Lefschetz fibrations, complex structures and Seifert fibrations on S^1 X M^3
我们考虑乘积4-流形S^1XM，其中M是一个闭的、连通的、定向的3-流形。我们证明了如果S^1 × M允许一个复结构或一个Lefschetz纤维化或Seifert纤维化，则下列陈述为真：S^1 × M允许一个辛结构当且仅当M纤维在S^1上，在附加的假设下，M没有假3-胞腔。我们还讨论了S ^1 × M上M的几何结构和复结构与Seifert纤维化之间的关系。
We show that if a contact open book $(Σ,h)$ on a $(2n+1)$-manifold $M$ ($n\geq1$) is induced by a Lefschetz fibration $π:W \to D^2$, then there is a one-to-one correspondence between positive stabilizations of $(Σ,h)$ and \emph{positive stabilizations} of $π$. More precisely, any positive stabilization of $(Σ,h)$ is induced by the corresponding positive stabilization of $π$, and conversely any positive stabilization of $π$ induces the corresponding positive stabilization of $(Σ,h)$. We define \emph{exact open books} as boundary open books of compatible exact Lefschetz fibrations, and show that…
It is known that an arbitrary smooth, oriented 4-manifold admits the structure of what is called a broken Lefschetz fibration. Given a broken fibration, there are certain modifications, realized as homotopies of the fibration map, that enable one to construct infinitely many distinct fibrations of the same manifold. The aim of this paper is to prove that these modifications are sufficient to obtain every broken fibration in a given homotopy class of smooth maps. One notable application is that adding an additional "projection" move generates all broken fibrations, regardless of homotopy class.…
We construct two types of non-holomorphic Lefschetz fibrations over $S^2$ with $(-1)$-sections ---hence, they are fiber sum indecomposable--- by giving the corresponding positive relators. One type of the two does not satisfy the slope inequality (a necessary condition for a fibration to be holomorphic) and has a simply-connected total space, and the other has a total space that cannot admit any complex structure in the first place. These give an alternative existence proof for non-holomorphic Lefschetz pencils without Donaldson's theorem.
We describe a Lefschetz fibration of genus one on the disk cotangent bundle of any closed orientable surface S. As a corollary, we obtain an explicit genus one open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of S.
In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a sequence of rational blowdowns from its minimal resolution.
We construct a positive allowable Lefschetz fibration over the disk on any minimal weak symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity.
In this paper, we study compact convex Lefschetz fibrations on compact convex symplectic manifolds (i.e., Liouville domains) of dimension $2n+2$ which are introduced by Seidel and later also studied by McLean. By a result of Akbulut-Arikan, the open book on $\partial W$, which we call \emph{convex open book}, induced by a compact convex Lefschetz fibration on $W$ carries the contact structure induced by the convex symplectic structure (i.e., Liouville structure) on $W$. Here we show that, up to a Liouville homotopy and a deformation of compact convex Lefschetz fibrations on $W$, any simply con…
Branched covers and pencils on hyperelliptic Lefschetz fibrations
本文推广既有研究，构造了由显式单值分解得到的无穷族辛4维流形，证明其微分同胚于两个标准超椭圆Lefschetz纤维化的纤维和复曲面，且这类纤维化及其纤维和拥有与次数加倍程序所得不同的显式无穷Lefschetz铅笔族。
Minimal number of singular fibers in a Lefschetz fibration
在亏格为h的闭（黎曼）曲面上存在一个（相对最小的）亏格为g的Lefschetz纤维化，其中只有一个奇异纤维当且仅当g>2且h>1.奇异纤维可以选择为可约的或不可约的。其它结果是亏格至少为3的闭曲面上的每个Dehn扭是两个换位子的乘积，并且任何闭曲面上的Dehn扭都不等于单个换位子。
By applying the lantern relation substitutions to the positive relation of the genus two Lefschetz fibration over $\mathbb{S}^{2}$. We show that $K3\#2 \overline{\mathbb{CP}}{}^{2}$ can be rationally blown down along seven disjoint copies of the configuration $C_2$. We compute the Seiberg-Witten invariant of the resulting symplectic 4-manifolds, and show that they are symplectically minimal. We also investigate how these exotic smooth 4-manifolds constructed via lantern relation substitution method are fiber sum decomposable. Furthermore by considering all the possible decompositions for each …
Lefschetz fibrations with unbounded Euler class
研究了Lefschetz纤维化的有界上同调。如果一个Lefschetz纤维化有亏格至少为2的正则纤维，并且它至少有两个不同的消失圈，我们证明了它的Euler类是没有界的。作为一个结果，我们排除了存在的负弯曲度量的Lefschetz纤维与一个以上的奇异纤维。
Adapting a construction of D Salamon involving the U(1) vortex equations, we explore the properties of a Floer theory for 3-manifolds that fiber over S^1 which exhibits several parallels with monopole Floer homology, and in all likelihood coincides with it. The theory fits into a restricted analogue of a TQFT in which the cobordisms are required to be equipped with Lefschetz fibrations, and has connections to the dynamics of surface symplectomorphisms.
Let M denote the total space of a Lefschetz fibration, obtained by blowing up a Lefschetz pencil on an algebraic surface. We consider the n-fold fibre sum M(n), generalizing the construction of the elliptic surfaces E(n). For a Lefschetz pencil on a simply-connected minimal surface of general type we partially calculate the Seiberg-Witten invariants of the fibre sum M(n) using a formula of Morgan-Szabo-Taubes. As an application we derive an obstruction for self-diffeomorphisms of the boundary of the tubular neighbourhood of a general fibre in M(n) to extend over the complement of the neighbour…
In this article we study proper symplectic and iso-symplectic embeddings of $4$--manifolds in $6$--manifolds. We show that a closed orientable smooth $4$--manifold admitting a Lefschetz fibration over $\C P^1$ admits a symplectic embedding in the symplectic manifold $(\C P^1 \times \C P^1 \times \C P^1, ω_{pr}),$ where $ω_{pr}$ is the product symplectic form on $\C P^1 \times \C P^1 \times \C P^1.$ We also show that there exists a sub-critical Weinstein $6$--manifold in which all finite type Weinstein $4$--manifolds admit iso-symplectic embeddings.
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 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…
From Morse Functions to Lefschetz Fibrations on Cotangent Bundles
本文证明紧流形上任意 Morse 函数与适配梯度可构造余切丛上同伦唯一的复值辛 Lefschetz 纤维化，并研究其正则纤维的拓扑与辛几何。
On hyperelliptic $C^\infty$-Lefschetz fibrations of four-manifolds
我们发现，超椭圆辛Lefschetz纤维化是辛双有理的两重覆盖的理性直纹面，分支在一个辛嵌入表面。这就把亏格2纤维化的分类简化为有理直纹曲面中某些辛子流形的分类。
Thanks to a result of Lisca and Matic and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic Spinc structures induce contact structures with distinct Ozsvath-Szabo invariants. Here we give an infinite family of examples showing that converse of Lisca-Matic-Plamenevskaya theorem does not hold in general. Our examples arise from Mazur type corks.
Loi-Piergallini and Akbulut-Ozbagci showed that every compact Stein surface admits a Lefschetz fibration over the 2-disk with bounded fibers. In this note we give a more intrinsic alternative proof of this result.
Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality
证明了半稳定辛Lefschetz纤维化中临界点个数的上界。我们还得到了Lefschetz束中非分离消失圈个数的一个新的下界，并重新证明了已知的Dehn扭的换位子长度的下界.
Symplectic submanifolds in dimension $6$ from hyperelliptic Lefschetz fibrations
本研究利用4-流形上的超椭圆Lefschetz纤维化，构造出含无穷多同调等价但同伦不等价且无复结构的余维2辛子流形的闭单连通辛6-流形，并推广至高维，还在附录证明了ℂℙ³中低次余维2辛子流形的微分同胚性。
We describe a construction of the Fukaya category of an exact symplectic Lefschetz fibration, together with its closed-open string map.
Nearly Lefschetz fibrations, Stein structures, and regular Lagrangians
研究近Lefschetz纤维化上的Stein结构，证明四维Weinstein域中具Legendrian边界的Lagrangian圆盘均为正则，解决邻近猜想与Slice-Ribbon猜想的Lagrangian类比，并给出构造Weinstein Kirby图的算法。
Positive Dehn Twist Expressions for some New Involutions in Mapping Class Group
众所周知的事实是，任何亏格$g$辛莱夫谢茨纤维化$ X^{4}\to S^{2}$是由一个等于映射类群中单位元的字给出的，并且其每个元素都由正的Dehn扭曲给出，这提供了映射类群中的字与被实现为辛莱夫谢茨纤维化的4-流形之间的密切关系。本文在映射类群中引入新的概念，从而得到新的辛Lefschetz纤维化。我们还计算这些辛Lefschetz纤维化的签名。
Lipshitz, Ozsváth and Thurston defined a bordered Heegaard Floer invariant CFDA for 3-manifolds with two boundary components, including mapping cylinders for surface diffeomorphisms. We define a related invariant for certain 4-dimensional cobordisms with corners, by associating a morphism F from CFDA(f) to CFDA(g) to each such cobordism between two mapping cylinders f and g. Like the Ozsváth-Szabó invariants of cobordisms between closed 3-manifolds, this morphism arises from counting holomorphic triangles on Heegaard triples. We demonstrate that the homotopy class of the morphism F only depend…
We (re)consider how the Fukaya category of a Lefschetz fibration is related to that of the fibre. The distinguishing feature of the approach here is a more direct identification of the bimodule homomorphism involved.
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