Lefschetz Fibrations and 3-Fold Branched Covering Spaces

A note on sections of broken Lefschetz fibrations

We show that there exists a non-trivial simplified broken Lefschetz fibration which has infinitely many homotopy classes of sections. We also construct a non-trivial simplified broken Lefschetz fibration which has a section with non-negative square. It is known that no Lefschetz fibration satisfies either of the above conditions. Smith proved that every Lefschetz fibration has only finitely many homotopy classes of sections, and Smith and Stipsicz independently proved that a Lefschetz fibration is trivial if it has a section with non-negative square. So our results indicate that there are no g…

关于莱夫谢茨纤维化拓扑的注记

A note on the topology of Lefschetz fibrations

该研究给出非平凡亏格g莱夫谢茨纤维化第一贝蒂数的上界，证明单值化传递时莱夫谢茨纤维化单连通，并讨论了一类具极大非平凡第一同调的不可分解亏格2莱夫谢茨纤维化的潜在应用。

On genus-1 simplified broken Lefschetz fibrations

Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism types of such fibrations with connected fibers and with less than six Lefschetz singularities. In these studies, we obtain several families of genus-1 simplified broken Lefschetz fibrations, which we conjecture contain all such fibrations, and determine the diffeomorphism types of th…

Signatures of Lefschetz fibrations

Let M be a smooth 4-manifold which admits a genus g Lefschetz fibration over D^2 or S^2. We develop a technique to compute the signature of M using the global monodromy of this fibration.

不可定向Lefschetz纤维化的定向双复盖

Orientation double covers of non-orientable Lefschetz fibrations

该研究证明标准定向双复盖与不可定向Lefschetz纤维化的复合为非手性Lefschetz纤维化并给出其单值分解，还提出三种不改变该类纤维化同构类的单值分解变换。

Charts, signatures, and stabilizations of Lefschetz fibrations

We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz fibrations over the 2-sphere obtained by Endo and Nagami to that for Lefschetz fibrations over arbitrary closed oriented surface. We then show two theorems on stabilization of Lefschetz fibrations under fiber summing with copies of a typical Lefschetz fibration as generalizations…

A real open book not fillable by a real Lefschetz fibration

A real 3- or 4-manifold has by definition an orientation preserving smooth involution acting on it. We consider Lefschetz fibrations of 4-dimensional manifolds-with-boundary and open book decompositions on their boundary in the existence of a real structure. We prove that there is a real open book which cannot be filled by a real Lefschetz fibration, although it is filled by non-real Lefschetz fibrations.

Geography of symplectic Lefschetz fibrations and rational blowdowns

We produce simply connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides a symplectic extension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blow…

Genus-$3$ Lefschetz Fibrations and Exotic $4$-Manifolds with $b_{2}^{+}=3$

We explicitly construct a genus-$3$ Lefschetz fibration over $\mathbb{S}^{2}$ whose total space is $\mathbb{T}^{2}\times \mathbb{S}^{2}\# 6\overline{\mathbb{C} P^{2}}$ using the monodromy of Matsumoto's genus-$2$ Lefschetz fibration. We then construct more genus-$3$ Lefschetz fibrations whose total spaces are exotic minimal symplectic $4$-manifolds $3 \mathbb{C} P^{2} \# q\overline{\mathbb{C} P^{2}}$ for $q=13,\ldots,19$. We also generalize our construction to get genus-$3k$ Lefschetz fibration structure on the $4$-manifold $Σ_{k}\times \mathbb{S}^{2}\# 6\overline{\mathbb{C} P^{2}}$ using the …

A note on embedding of achiral Lefschetz fibrations

We discuss $4$-dimensional achiral Lefschetz fibrations bounding $3$-dimensional open books and study their Lefschetz fibration (LF) embedding in a bounded $6$-dimensional manifold, in the sense of Ghanwat--Pancholi. As an application we give another proof of the fact that every closed orientable $4$-manifold embeds in $S^2 \times S^2 \times S^2$ . We also show that every achiral Lefschetz fibration with hyperelliptic monodromy admits LF embedding in $D^6 = D^2 \times D^4$ and discuss an obstruction to such LF embeddings.

On sections of genus two Lefschetz fibrations

In this note we find new relations in the mapping class group of a genus two surface with n boundary components for n=1,..., 8 which induce a genus two Lefschetz fibration $CP^2#13CP^2bar \to S^2$ with n disjoint sections. As a consequence, we observe any holomorphic genus 2 Lefschetz fibration without separating singular fibers admits a section.

Four-manifolds admitting hyperelliptic broken Lefschetz fibrations

We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus $g\geq 3$ and extend these involutions to the four-manifolds obtained by blowing up the total spaces. The extended involutions induce double branched coverings over blown up sphere bundles over the sphere. We also show that the regular fiber of such a fibration of genus $g\geq 3$ represents a non-trivial rational homology class of the total space.

Topological uniqueness results for Lefschetz fibrations over the disc

We prove that a Lefschetz fibration over the disc that, after compactification, has the same singular fibers as an extremal rational elliptic surface can be obtained by deleting a singular fiber and a section from the rational extremal elliptic surface, i.e. such a Lefschetz fibration is determined up to topological equivalence by its set of singular fibers. We get a complete clasification of Lefschetz fibrations with 2 $I_1$ fibers as a byproduct of our results. The proof is inspired by homological mirror symmetry and Karpov--Nogin's theorem on constructivity of helices on del Pezzo surfaces.…

Broken Lefschetz fibrations and smooth structures on 4-manifolds

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.

Universal Lefschetz fibrations over bounded surfaces

In analogy with the vector bundle theory we define universal and strongly universal Lefschetz fibrations over bounded surfaces. After giving a characterization of these fibrations we construct very special strongly universal Lefschetz fibrations when the fiber is the torus or an orientable surface with connected boundary and the base surface is the disk. As a by-product we also get some immersion results for 4-dimensional 2-handlebodies.

The Number of Singular Fibers in Hyperelliptic Lefschetz Fibrations

We consider complex surfaces, viewed as smooth $4$-dimensional manifolds, that admit hyperelliptic Lefschetz fibrations over the $2$-sphere. In this paper, we show that the minimal number of singular fibers of such fibrations is equal to $2g+4$ for even $g\geq4$. For odd $g\geq7$, we show that the number is greater than or equal to $2g+6$. Moreover, we discuss the minimal number of singular fibers in all hyperelliptic Lefschetz fibrations over the $2$-sphere as well.

Lefschetz fibration structures on knot surgery 4-manifolds

In this article we study Lefschetz fibration structures on knot surgery 4-manifolds obtained from an elliptic surface E(2) using Kanenobu knots $K$. As a result, we get an infinite family of simply connected mutually diffeomorphic 4-manifolds coming from a pair of inequivalent Kanenobu knots. We also obtain an infinite family of simply connected symplectic 4-manifolds, each of which admits more than one inequivalent Lefschetz fibration structures of the same generic fiber.

Lefschetz fibrations and an invariant of finitely presented groups

Every finitely presented group is the fundamental group of the total space of a Lefschetz fibration. This follows from results of Gompf and Donaldson, and was also proved by Amoros-Bogomolov-Katzarkov-Pantev. We give another proof by providing the monodromy explicitly. We then define the genus of a finitely presented group $Γ$ to be the minimal genus of a Lefschetz fibration with fundamental group $Γ$. We also give some estimates of the genus of certain groups.

构造具有任意斜率的Lefschetz纤维

Constructing Lefschetz Fibrations with Arbitrary Slope

研究证明对于任意有理数r∈(2,8)，存在一个高 genus 的Lefschetz纤维，其斜率为r。

Minimal number of singular fibers in a nonorientable Lefschetz fibration

We show that there exists an admissible nonorientable genus $g$ Lefschetz fibration with only one singular fiber over a closed orientable surface of genus $h$ if and only if $g \geq 4$ and $h \geq 1$.

On sections of hyperelliptic Lefschetz fibrations

We construct a relation among right-handed Dehn twists in the mapping class group of a compact oriented surface of genus g with 4g+4 boundary components. This relation gives an explicit topological description of 4g+4 disjoint (-1)-sections of a hyperelliptic Lefschetz fibration of genus g on the manifold {CP}^2#(4g+5){-CP}^2.

Small Lefschetz Fibrations on Simply-Connected $4$-Manifolds

We consider simply-connected $4$-manifolds admitting Lefschetz fibrations over the $2$-sphere. We explicitly construct nonhyperelliptic and hyperelliptic Lefschetz fibrations of genus $4$ on simply-connected $4$-manifolds which are exotic symplectic $4$-manifolds in the homeomorphism classes of $\mathbb{C}P^{2}\#8\overline{\mathbb{C} P^{2}}$ and $\mathbb{C} P^{2}\#9\overline{\mathbb{C} P^{2}}$, respectively. From these, we provide upper bounds for the minimal number of singular fibers of such fibrations. In addition, we prove that this number is equal to $18$ for $g=3$ when such fibrations are…

Lefschetz fibrations on knot surgery $4$-manifolds via Stallings twist

In this article we construct a family of knot surgery $4$-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface $E(2)$ using connected sums of fibered knots obtained by Stallings twist from a slice knot $3_1 \sharp 3^*_1$. By comparing their monodromy groups induced from the corresponding monodromy factorizations, we show that they admit mutually nonisomorphic Lefschetz fibration structures.

On genera of Lefschetz fibrations and finitely presented groups

It is known that every finitely presented group is the fundamental group of the total space of a Lefschetz fibration. In this paper, we give another proof which improves the result of Korkmaz. In addition, Korkmaz defined the genus of a finitely presented group. We also evaluate upper bounds for genera of some finitely presented groups.

Partial fiber sum decompositions and signatures of Lefschetz fibrations

In his Ph.D. thesis, Burak Ozbagci described an algorithm computing signatures of Lefschetz fibrations where the input is a factorization of the monodromy into a product of Dehn twists. In this note, we give a reformulation of Ozbagci's algorithm which becomes much easier to implement. Our main tool is Wall's non-additivity formula applied to what we call partial fiber sum decomposition of a Lefschetz fibration over 2-disk. We show that our algorithm works for bordered Lefschetz fibrations over disk and it yields a formula for the signature of branched covers where the branched loci are regula…

非手性 Lefschetz 纤维化与曲线模空间

Achiral Lefschetz fibrations and the moduli space of curves

本文利用 Riemann 几何证明非手性 Lefschetz 纤维化存在到曲线模空间 Deligne–Mumford 紧化的分类映射，并将 Smith 符号差公式推广至非手性情形。

Counting Dirac braid relators and hyperelliptic Lefschetz fibrations

We define a new invariant $w$ for hyperelliptic Lefschetz fibrations over closed oriented surfaces, which counts the number of Dirac braids included intrinsically in the monodromy, by using chart description introduced by the second author. As an application, we prove that two hyperelliptic Lefschetz fibrations of genus $g$ over a given base space are stably isomorphic if and only if they have the same numbers of singular fibers of each type and they have the same value of $w$ if $g$ is odd. We also give examples of pair of hyperelliptic Lefschetz fibrations with the same numbers of singular f…

Upper bounds for the minimal number of singular fibers in a Lefschetz fibration over the torus

In this paper, we give some relations in the mapping class groups of oriented closed surfaces in the form that a product of a small number of right hand Dehn twists is equal to a single commutator. Consequently, we find upper bounds for the minimal number of singular fibers in a Lefschetz fibration over the torus.

Classification of totally real elliptic Lefschetz fibrations via necklace diagrams

We show that totally real elliptic Lefschetz fibrations that admit a real section are classified by their "real loci" which is nothing but an $S^1$-valued Morse function on the real part of the total space. We assign to each such real locus a certain combinatorial object that we call a \emph{necklace diagram}. On the one hand, each necklace diagram corresponds to an isomorphism class of a totally real elliptic Lefschetz fibration that admits a real section, and on the other hand, it refers to a decomposition of the identity into a product of certain matrices in $PSL(2,\Z)$. Using an algorithm …

On the minimal number of singular fibers in Lefschetz fibrations over the torus

We show that the minimal number of singular fibers $N(g,1)$ in a genus-$g$ Lefschetz fibration over the torus is at least $3$. As an application, we show that $N(g, 1) \in \{ 3, 4\}$ for $g\ge 5$, $N(g, 1) \in \{3, 4,5 \}$ for $g= 3, 4$ and $N(2,1) = 7$.

Lefschetz fibrations over the disc

We provide a complete set of moves relating any two Lefschetz fibrations over the disk having as their total space the same 4-dimensional 2-handlebody up to 2-equivalence. As a consequence, we also obtain moves relating diffeomorphic 3-dimensional open books, providing a different approach to an analogous previous result by Harer.

Simplified broken Lefschetz fibrations and trisections of 4-manifolds

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.

A note on signature of Lefschetz fibrations with planar fiber

Using theorems of Eliashberg and McDuff, Etnyre [Et] proved that the intersection form of a symplectic filling of a contact 3-manifold supported by planar open book is negative definite. In this paper, we prove a signature formula for allowable Lefschetz fibrations over $D^2$ with planar fiber by computing Maslov index appearing in Wall's non-additivity formula. The signature formula leads to an alternative proof of Etnyre's theorem via works of Niederkrüger and Wendl [NWe] and Wendl [We]. Conversely, Etnyre's theorem, together with the existence theorem of Stein structures on Lefschetz fibrat…

4-流形的（非手性）Lefschetz纤维化嵌入

(Achiral) Lefschetz fibration embeddings of $4$-manifolds

本文证明了紧致连通可定向4-流形的多种Lefschetz纤维化嵌入定理，推广至$\mathbb CP^1$并给出闭流形在$\mathbb CP^2\times\mathbb CP^1$中的嵌入，由此给出Hirsch定理的新证明。

Quandles and Lefschetz Fibrations

We show that isotopy classes of simple closed curves in any oriented surface admit a quandle structure with operations induced by Dehn twists, the Dehn quandle of the surface. We further show that the monodromy of a Lefschetz fibration can be conveniently encoded as a quandle homomorphism from the knot quandle of the base as a manifold with a codimension 2 subspace (the set of singular values) to the Dehn quandle of the generic fibre, and discuss prospects for construction of invariants arising naturally from this description of the monodromy.

关于Lefschetz纤维化和2复形上纤维 bundle 的截面

On sections of Lefschetz fibrations and bundles over 2-complexes

研究了纤维化截面存在的条件，证明了有限2复形上的纤维 bundle 截面存在的充要条件，并给出了Lefschetz纤维化截面的代数判据及应用。

通过 Lefschetz 纤维化构造 3-流形群在 PU(2,1) 中的表示

A construction of representations of 3-manifold groups into PU(2,1) through Lefschetz fibrations

本文通过 Lefschetz 纤维化构造了 3-流形基本群在复双曲空间全纯等距群格点中的无穷多个非共轭表示，该格点对应于 Hirzebruch 超平面布置的分支覆盖配边基本群。

Constructing broken Lefschetz fibrations from handle decompositions

We present an approach to constructing broken Lefschetz fibrations (BLFs) $f:X\rightarrow S^2$ from a handle decomposition of a 4-manifold $X$. Given a handle decomposition as input these techniques yield explicit descriptions of the BLFs, and do not rely on classification results from contact topology or the choice of a generic indefinite function like some earlier approaches. We show that this approach will always work in the case when $X$ is the double of a 4-manifold with boundary, and compute explicit examples of BLFs on connected sums of $S^2$-bundles.

Bilinear-form invariants of Lefschetz fibrations over the 2-sphere

We introduce invariants of Hurwitz equivalence classes with respect to arbitrary group $G$. The invariants are constructed from any right $G$-modules $M$ and any $G$-invariant bilinear function on $M$, and are of bilinear forms. For instance, when $G$ is the mapping class group of the closed surface, $\mathcal{M}_g$, we get an invariant of 4-dimensional Lefschetz fibrations over the 2-sphere. Moreover, the construction is applicable for the quantum representations of $\mathcal{M}_g $ derived from Chern-Simons field theory. We compute the associated invariants in some cases, and find infinitely…

On the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4

A complete description of the global monodromy of a Lefschetz fibration arising from the Fermat surface of degree 4 is given. As a by-product we get a positive relation among right hand Dehn twists in the mapping class group of a closed orientable surface of genus 3.

Fiber sums of genus 2 Lefschetz fibrations

Using the recent results of Siebert and Tian about the holomorphicity of genus 2 Lefschetz fibrations with irreducible singular fibers, we show that any genus 2 Lefschetz fibration becomes holomorphic after fiber sum with a holomorphic fibration.

Cobordism maps on periodic Floer homology induced by elementary Lefschetz fibrations

Periodic Floer homology (PFH) is a Gromov-Floer type invariant for fibered three-manifolds with Hamiltonian structures. The cobordism maps on periodic Floer homology induced by symplectic cobordisms are currently only defined indirectly by using Seiberg-Witten theory. In this paper, we investigate the cobordism maps induced by a class of symplectic cobordisms constructed by P. Seidel, called elementary Lefschetz brations. We define the PFH cobordism maps induced by elementary Lefschetz fibrations in terms of holomorphic curves. Moreover, we compute these maps for some cases.

Genus two Lefschetz fibrations with $b^{+}_{2}=1$ and ${c_1}^{2}=1,2$

In this article we construct a family of genus two Lefschetz fibrations $f_{n}: X_{θ_n} \rightarrow \mathbb{S}^{2}$ with $e(X_{θ_n})=11$, $b^{+}_{2}(X_{θ_n})=1$, and $c_1^{2}(X_{θ_n})=1$ by applying a single lantern substitution to the twisted fiber sums of Matsumoto's genus two Lefschetz fibration over $\mathbb{S}^2$. Moreover, we compute the fundamental group of $X_{θ_n}$ and show that it is isomorphic to the trivial group if $n = -3$ or $-1$, $\mathbb{Z}$ if $n =-2$, and $\mathbb{Z}_{|n+2|}$ for all integers $n\neq -3, -2, -1$. Also, we prove that our fibrations admit $-2$ section, show tha…

Classification of broken Lefschetz fibrations with small fiber genera

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.

A genus zero Lefschetz fibration on the Akbulut cork

We first construct a genus zero positive allowable Lefschetz fibration over the disk (a genus zero PALF for short) on the Akbulut cork and describe the monodromy as a positive factorization in the mapping class group of a surface of genus zero with five boundary components. We then construct genus zero PALFs on infinitely many exotic pairs of compact Stein surfaces such that one is a cork twist of the other along an Akbulut cork. The difference of smooth structures on each of exotic pairs of compact Stein surface is interpreted as the difference of the corresponding positive factorizations in …

Simplifying indefinite fibrations on 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. 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 lo…

Lefschetz fibrations on nonorientable 4-manifolds

Let $W$ be a nonorientable $4$-dimensional handlebody without $3$- and $4$-handles. We show that $W$ admits a Lefschetz fibration over the $2$-disk, whose regular fiber is a nonorientable surface with nonempty boundary. This is an analogue of a result of Harer obtained in the orientable case. As a corollary, we obtain a $4$-dimensional proof of the fact that every nonorientable closed $3$-manifold admits an open book decomposition, which was first proved by Berstein and Edmonds using branched coverings. Moreover, the monodromy of the open book we obtain for a given $3$-manifold belongs to the …

Hyperelliptic Lefschetz fibrations and branched covering spaces

Let M be a smooth 4-manifold which admits a relatively minimal hyperelliptic genus h Lefschetz fibration over the 2-sphere. If all of the vanishing cycles for this fibration are nonseparating curves, then we show that M is a 2-fold cover of a 2-sphere bundle over the 2-sphere, branched over an embedded surface. If the collection of vanishing cycles for this fibration includes separating curves, we show that M is the relative minimalization of a Lefshetz fibration N, with N a 2-fold branched cover of a rational surface, branched over an embedded surface.

Sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration

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.

Noncomplex smooth 4-manifolds with genus-2 Lefschetz fibrations

We construct noncomplex smooth 4-manifolds which admit genus-2 Lefschetz fibrations over S^2. The fibrations are necessarily hyperelliptic, and the resulting 4-manifolds are not even homotopy equivalent to complex surfaces. Furthermore, these examples show that fiber sums of holomorphic Lefschetz fibrations do not necessarily admit complex structures.

具有柄体单演（handlebody monodromy）的 Lefschetz 纤维化

Lefschetz fibrations with handlebody monodromy

本文研究柄体单演的相对极小 Lefschetz 纤维化，证明其符号差等于负临界点数，并刻画存在性条件及超椭圆情形与换位子长度性质。

The existence of an indecomposable minimal genus two Lefschetz fibration

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.

Lefschetz pencils, branched covers and symplectic invariants

This set of lectures aims to give an overview of Donaldson's theory of linear systems on symplectic manifolds and the algebraic and geometric invariants to which they give rise. After collecting some of the relevant background, we discuss topological, algebraic and symplectic viewpoints on Lefschetz pencils and branched covers of the projective plane. The later lectures discuss invariants obtained by combining this theory with pseudo-holomorphic curve methods.

Minimality and fiber sum decompositions of Lefschetz fibrations

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.