R. Inanc Baykur, Mustafa Korkmaz
We construct symplectic surface bundles over surfaces with positive signatures for all but 18 possible pairs of fiber and base genera. Meanwhile, we determine the commutator lengths of a few new mapping classes.
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The nonorientable 4-genus is an invariant of knots which has been studied by many authors, including Gilmer and Livingston, Batson, and Ozsváth, Stipsicz, and Szabó. Given a nonorientable surface $F \subset B^4$ with $\partial F = K\subset S^3$ a knot, an analysis of the existing methods for bounding and computing the nonorientable 4-genus reveals relationships between the first Betti number $β_1$ of $F$ and the normal Euler class $e$ of $F$. This relationship yields a geography problem: given a knot $K$, what is the set of realizable pairs $(e(F), β_1(F))$ where $F\subset B^4$ is a nonorienta…
Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds
该研究基于Roulleau与Urzúa的复曲面成果，构造了自旋辛4-流形的无限多互不同胚光滑结构，并指出其结果接近辛Bogomolov-Miyaoka-Yau不等式给出的最优边界。
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…
On the geography of simply connected nonspin symplectic $4$-manifolds with nonnegative signature
在\cite{AP3, AHP}中，第一作者及其合作者构造了对于每个整数$n \geq 25$同胚但不同胚于$(2n-1){\mathbb{CP}}^{2}\#(2n-1)\overline{\mathbb{CP}}^{2}$的不可约辛$4$ -流形，以及具有正签名的单连通不可约非自旋辛$4$ -流形族，这些流形对于辛地理问题是有趣的。本文改进了\cite{AP3, AHP}中的主要结果。特别地，我们构造了（i）对于每个整数$n \geq 12$，无穷多个与$(2n-1){\mathbb{CP}^{2}}\#(2n-1)\overline{\mathbb{CP}}^{2}$同胚但不同胚的不可约辛和非辛$4$ -流形，和（ii）在所有已知的单连通$4$中具有最小欧拉特征的单连通不可约非自旋辛$4$ -流形族。流形与积极的签名和一个以上的光滑结构。我们的构造使用了在Bogomolov-Miyaoka-Yau线上的Hirzebruch和Bauer-Catanese的复杂曲面，以及在\cite{A4, AP1, ABBKP, AP2, AS}中构造的奇异辛$4$ -流形。
Surface Bundles With Non-Zero Signature
在本文中，我们开发了一种新的技术，产生无限多的表面丛与非零签名。
Flat structures on surface bundles
我们证明了存在具有非平凡法丛的闭叶平面丛。这使我们能够计算具有标记点的曲面单同态群的阿贝尔化。我们还延长坪井的公式，表示欧拉类的平坦的圆丛的Calabi不变的某些Hamilton同构的表面更高的属，并推导出类似的第一个MMM类的表面丛穿孔纤维。
We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism.
In 1998, Gompf described a Stein domain structure on the disk cotangent bundle of any closed surface S, by a Legendrian handlebody diagram. We prove that Gompf's Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorphic to the unit cotangent bundle of S equipped with its canonical contact structure. As a corollary, we obtain a surgery diagram for the canonical contact structure on the unit cotangent bundle of S.
We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, namely the ones that arise through a ramified covering. These are the main known examples of bundles with non-zero signature.
Commensurability of Surface Automorphisms \\and surface bundles over $S^1$
这篇笔记将成为一篇有更多作者的新论文的一部分。
Indecomposable surface bundles over surfaces
对任意一对整数g至少为2，h至少为1，我们在亏格h曲面上构造了无穷多个纤维和与截和不可分解的亏格g曲面丛，它们的全空间是两两同伦不等价的.
Mapping class groups of Heegaard splittings of surface bundles
每个亏格为$g$纤维的表面丛都有亏格为$2g+1$的典型Heegaard分裂.在曲面丛具有足够复杂的单值映射的情况下，我们对这类Heegaard分裂的映射类群进行了分类.
In the present note we describe geometrically the homology classes in the total space of a surface bundle over a surface in terms of the holonomy map. We treat the cases where the base surface is closed or has one boundary component. We replace the wrong theorem of the previous version by some observations on the homology of surface bundles and explain why our previous strategy failed.
In this paper we give the first example of a surface bundle over a surface with at least three fiberings. In fact, for each $n \ge 3$ we construct $4$-manifolds $E$ admitting at least $n$ distinct fiberings $p_i: E \to Σ_{g_i}$ as a surface bundle over a surface with base and fiber both closed surfaces of negative Euler characteristic. We give examples of surface bundles admitting multiple fiberings for which the monodromy representation has image in the Torelli group, showing the necessity of all of the assumptions made in the main theorem of our recent paper [arXiv:1404.0066]. Our examples s…
Surface bundles with genus two Heegaard splittings
已知存在亏格为任意高的曲面丛，它们具有亏格为2的Heegaard分裂。最简单的例子是塞弗特纤维空间，它以球体为底空间，有三根特殊的纤维，并允许有水平表面。我们的特点是monodromy地图的所有表面束属两个Heegaard分裂，并表明，每一个是积分Dehn手术的结果，在这些塞弗特纤维空间沿环路的Heegaard表面相交的水平面。(This一种类型的手术保留了束结构和Heegaard分裂。
For a closed manifold $M$, let Fib$(M)$ be the number of distinct fiberings of $M$ as a fiber bundle with fiber a closed surface. In this paper we give the first computation of Fib$(M)$ where $1<\text{Fib}(M)<\infty$ but $M$ is not a product. In particular, we prove Fib$(M)=2$ for the Atiyah-Kodaira manifold and any finite cover of a trivial surface bundle. We also give an example where Fib$(M)=4$.
The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus $g$ surface bundles as branched double covers of the $3$-sphere behaves like 1/$g$. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed $3$-manifolds are branched double covers of the $3…
The signature of a surface bundle over a surface is known to be divisible by 4. It is also known that the signature vanishes if the fiber genus is less than or equal to 2 or the base genus is less than or equal to 1. In this article, we construct new smooth 4-manifolds with signature 4 which are surface bundles over surfaces with small fiber and base genera. From these we derive improved upper bounds for the minimal genus of surfaces representing the second homology classes of a mapping class group.
Turaev--Viro invariants and profinite completions of surface bundles
本文在曲面群 pro-同余共轭可分且基本群正则有限完备同构的条件下，证明两个圆周上的曲面丛对所有球融合范畴具有相同的 Turaev–Viro 不变量。
Extending a result of Morita, we show that all tautological classes of the moduli space of genus g curves are bounded. As an application, we obtain that for a surface bundle $E\to B$ over a closed surface, the Eulder characteristic $χ(E)$ and the signature $σ(E)$ are related by $\vert 3 σ(E)\vert \leq \vert χ(E)\vert$.
We classify incompressible, boundary-incompressible, nonorientable surfaces in punctured-torus bundles over $S^1$. We use the ideas of Floyd, Hatcher, and Thurston. The main tool is to put our surface in the "Morse position" with respect to the projection of the bundle into the basis S^1.
Multisections of surface bundles and bundles over the circle
本文证明曲面丛和纤维已多剖分的圆上纤维丛在任意维度均承认多剖分，并给出显式构造与示例。
We describe the second integral cohomology group of a surface bundle as the group of Chern classes of fiberwise holomorphic complex line bundles and use this to obtain information on this group.
On the divisibility of characteristic classes of non-oriented surface bundles
本文介绍了一种构造方法，它赋予任意流形丛以纤维方向覆盖。这是用来表明，zeta类的无取向的表面束是不可分的稳定范围内。
Complex Hyperbolic Structures on Disc Bundles over Surfaces. II. Example of a Trivial Bundle
本文基于[AGG]中开发的方法。我们在一个闭的可定向曲面$Σ$（亏格2）上的平凡圆盘丛上构造了一个复双曲结构，从而解决了复双曲几何中一个长期存在的问题（见[Gol1，p.583]和[Sch，p.14]）.这个例子也回答了[Eli，开放问题8.1]，如果亏格>1的闭曲面上的平凡圆丛允许全纯可填充的切触结构。构造的例子M满足关系$2(χ+e)=3τ$，这是存在丛的全纯截面所必需的，其中$χ=χΣ$代表$Σ$的欧拉示性数，e=eM，表示丛的欧拉数，$τ$表示托莱多不变量。(The这个关系对于[AGG]中构造的一系列例子也是有效的。）开放问题：是否存在丛M的全纯截面？
The Euler class of infinite-type surface bundles
本文研究带截面的无限型曲面丛的欧拉类，证明其非平凡性及幂次行为依赖亏格与端点类型，并将Morita定理推广至无限亏格曲面。
Pseudo-Anosov extensions and degree one maps between hyperbolic surface bundles
设$F',F$是亏格$g'>g\ge 1$的任意两个闭可定向曲面，$f:F\to F$是任意伪Anosov映射.然后将$f$“扩展”为一个伪Anosov映射$f':F'\to F'$，使得双曲曲面丛之间存在一个保纤维度为1的映射$M(F',f')\to M(F,f)$.此外，可以选择扩展$f'$，使得表面丛$M(F',f')$和$M(F,f)$具有相同的第一Betti数。
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