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. 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 near-symplectic 4-manifold.
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 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…
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.
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.
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.
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.
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 universal Lefschetz fibrations, defined in analogy with classical universal bundles. We also introduce the cobordism groups of Lefschetz fibrations, and we see how these groups are quotients of the singular bordism groups via the universal Lefschetz fibrations.
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.
在本文中，我们介绍了真实莱夫谢茨纤维化的某些不变量。我们称这些不变量为{ \em real Lefschetz chains}。证明了当纤维亏格大于1时，实Lefschetz链是具有实临界值的实Lefschetz纤维化的完全不变量.然而，如果纤维亏格是1，真正的莱夫谢茨链是不足以区分真正的莱夫谢茨纤维。我们表明，通过添加一定的二元装饰真正的Lefschetz链，我们得到一个完整的不变量。
For $g\geq 3$, we construct genus-$g$ Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to $2$. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that the infimum and the supremum of slopes all Lefschetz fibrations are not realized as slopes.
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.
