具有非零签名的曲面束

Signatures of surface bundles and stable commutator lengths of Dehn twists

The first aim of this paper is to give four types of examples of surface bundles over surfaces with non-zero signature. The first example is with base genus 2, a prescribed signature, a 0-section and the fiber genus greater than a certain number which depends on the signature. This provides a new upper bound on the minimal base genus for fixed signature and fiber genus. The second one gives a new asymptotic upper bound for this number in the case that fiber genus is odd. The third one has a small Euler characteristic. The last is a non-holomorphic example. The second aim is to improve upper bo…

Geography of surface bundles over surfaces

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.

一个纤维丛的签名是乘模4

The signature of a fibre bundle is multiplicative mod 4

我们将一个闭的、定向的、$4k$维$PL$流形的模4签名表示为它的欧拉示性数和Korzeniewski [11]中定义的新的绝对挠率不变量的线性组合。设$F \to E \to B$是一个$PL$纤维丛，其中$F$、$E$和$B$是闭的、连通的、相容定向的$PL$流形。我们给出了全空间$E$的绝对挠率的一个公式，并结合这两个结果证明了$E$的签名与$F$和$B$的签名之积模4全等.

The non-multiplicativity of the signature modulo 8 of a fibre bundle is an Arf-Kervaire invariant

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group of the base acts trivially on the cohomology ring of the fibre with real coefficients, in which case the signature of the total space equals the product of the signatures of base and fibre. Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo 4. In this paper we present two results concerning the multiplicativity modulo 8: firstly we identify the obstruction to multiplicativity modulo 8 with the Arf-Kervaire invariant of a Po…

纤维丛的模8签名

The signature modulo 8 of fibre bundles

本文研究了4k维定向几何庞加莱复形的模4和模8的剩余数。利用Z4值二次型的Z 8值Brown-Kervaire不变量，证明了若签名可被4整除，则可被8整除的性质可由Z2值二次型的Arf不变量来判定.这个结果被应用到纤维化的几何庞加莱复体与全空间4k维。已知签名是乘法模4，并且存在非乘法模8的示例。通过将Z_4值Pontryagin平方除以2，我们确定了在适当的Z_2上同调向量空间上定义的Z_2值二次型的Arf不变量对模8乘性的阻碍。在一定的假设下，障碍物消失。

紧致曲面上平坦酉丛的符号数的可能值

On possible values of the signature of flat unitary bundles over compact surfaces

本文确定了紧致有向正亏格带边曲面上平坦 Hermite 丛的符号数范围，并证明所有满足给定不等式的整数均可由块对角表示实现。

带边曲面上平坦酉丛的符号差

Signature for flat unitary bundles over surfaces with boundary

本文研究带边紧曲面基本群在Hermitian型经典单Lie群中的表示，通过扩展Atiyah-Patodi-Singer的rho不变量，将符号差、Toledo与rho不变量以统一方式呈现，给出带边流体经典符号差公式的一个版本。

Kodaira维数为零的辛4-流形的四元数丛与贝蒂数

Quaternionic Bundles and Betti Numbers of Symplectic 4-Manifolds with Kodaira Dimension Zero

Kodaira维数为零的辛4-流形可以看作是辛卡-丘曲面。通过证明四元数向量丛的两个一般结果，我们能够完全确定它们的贝蒂数.

Double Kodaira fibrations with small signature

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fibre bundles. They are known to have positive signature divisible by $4$. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyse by studying the monodromy action for bundles of punctured curves. As a by-product we obtain a classification of all fix-point-free automorphisms on curves of genus at most $9$.

Signature Cocycles on the Mapping Class Group and Symplectic Groups

Werner Meyer constructed a cocycle in $H^2(Sp(2g, \mathbb{Z}); \mathbb{Z})$ which computes the signature of a closed oriented surface bundle over a surface, with fibre a surface of genus g. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study the signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise re…

非全纯曲面丛与Lefschetz纤维化

Non-holomorphic surface bundles and Lefschetz fibrations

我们证明了某些稳定化如何产生无穷多个闭的定向4-流形，这些流形是亏格g曲面丛的全空间。Lefschetz纤维化）在亏格h曲面上，并且具有非零特征，但不允许具有任何一个方向的复结构，对于“大多数”（分别为所有）可能的值，g至少为3，h至少为2（分别地，g至少为2，h非负）。

小亏格曲面上的曲面丛

Surface bundles over surfaces of small genus

我们构造了亏格为2的曲线上的光滑复射影曲线的非等平凡代数族的例子。这解决了一个问题，从柯比的问题列表中的低维拓扑。也就是说，我们表明，2是最小的可能的基亏格，可以出现在一个4-流形的非零签名，这是一个定向纤维丛的黎曼曲面。该问题的一个改进版本要求固定签名和纤维亏格的最小基亏格。我们的建设也提供了新的（渐近）上界这些数字。

具有纤维化边界流形族的Hodge和特征定理

Hodge and signature theorems for a family of manifolds with fibration boundary

设$\bar{M}$是一个流形，其边界$Y$是一个纤维丛的全空间，并由一个边界定义函数$x$的消失来定义。本文证明了$L^2$ Hodge定理和$M$的签名定理，该$M$具有$dx^2 + x^{2c} h + k$形式的度量，其中$k$是基于纤维丛的度量到$Y$的提升，$h$是$Y$上的一个2形式，它限制于每个纤维上的一个度量，而$0 \leq c \leq 1$ .这些度量在当$c=0$时的情况和当$c=1$时的情况之间插值，在这种情况下，边界附近的度量是圆柱体，在这种情况下，边界附近的度量是边界纤维化的基部上的锥束。我们表明，$L^2$霍奇定理的上同调给出的最大和最小的扩展$d$关于这些度量和$L^2$签名定理的图像的最小上同调的最大上同调插值之间的已知结果为$L^2$霍奇和签名定理的圆柱和锥束型度量。特别地，霍奇定理都将$L^2$调和形式的相关空间与$X$的变化Peru的交上同调联系起来，$X$是由$\bar{M}$通过在边界处折叠$Y$的纤维而形成的空间。签名定理涉及戴所描述的$τ$不变量的变化。

零轨迹的次解析丛和管状邻域

Subanalytic Bundles and Tubular Neighbourhoods of Zero-Loci

我们引入了实解析流形$M$的子解析子集$X$上的子解析丛（具有有限维向量空间$P$的截面）的自然和相当一般的概念，并证明了当$M$是紧的时，在$P$中存在Baire截面子集$U$，其零轨迹在$X$中具有管状邻域，同胚于给定丛对这些零轨迹的限制.

零特征4-流形上的奇异光滑结构

Exotic smooth structures on 4-manifolds with zero signature

对任意整数$k\geq 2$，我们在拓扑$4$ -流形$(2k-1)(S^2\times S^2)$和$(2k-1)(\CP#\CPb)$上构造了无限族互不同构的不可约光滑结构，它们是$S^2\times S^2$和$\CP#\CPb$的$2k-1$拷贝的连通和.

Simplicial volume of fiber bundles with nonpositively curved fibers

We prove the simplicial volume of the total space of a smooth fiber bundle with fiber being an oriented closed connected (occ) manifold of nonpositive curvature and negative Ricci curvature over an occ manifold with a closed universal covering is zero. Furthermore, if the fiber is an occ negatively curved manifold with dimension more than $2$, the simplicial volume of the total space is zero if and only if the simplicial volume of the base space is zero.

Characteristic classes of fiberwise branched surface bundles via arithmetic

This paper is about cohomology of mapping class groups from the perspective of arithmetic groups. For a closed surface $S$ of genus $g$, the mapping class group $Mod(S)$ admits a well-known arithmetic quotient $Mod(S)\rightarrow Sp(2g, Z)$, under which the stable cohomology of $Sp(2g,Z)$ pulls back to algebra generated by the odd MMM classes of $Mod(S)$. We extend this example to other arithmetic groups associated to mapping class groups and explore some of the consequences for surface bundles. For $G=Z/mZ$ and for a regular $G$-cover $S\rightarrow S'$ (possibly branched), a finite index subgr…

Hirzebruch $χ_y$-genera modulo $8$ of fiber bundles for odd integers $y$

I. Hambleton, A. Korzeniewski and A. Ranicki proved that the signature of a fibre bundle of closed, connected, compatibly oriented PL manifolds is always multiplicative modulo 4. In this paper, we consider the Hirzebruch $χ_y$-genera for odd integers $y$ for a smooth fiber bundle such that the base, fibre, and total space are compact complex algebraic manifolds (in the complex analytic topology, not in the Zariski topology). We show that the Hirzebruch $χ_y$-genera of such a fibre bundle are always multiplicative modulo 4. We also investigate multiplicativity modulo 8 and show that if $y$ is c…

Geometric cycles and characteristic classes of manifold bundles

We produce new cohomology for non-uniform arithmetic lattices $Γ<SO(p,q)$ using a technique of Millson--Raghunathan. From this, we obtain new characteristic classes of manifold bundles with fiber a closed $4k$-dimensional manifold $M$ with indefinite intersection form of signature $(p,q)$. These classes are defined on a finite cover of $BDiff(M)$ and are shown to be nontrivial for $M=\#_g(S^{2k}\times S^{2k})$. In this case, the classes produced live in degree $g$ and are independent from the algebra generated by the stable (i.e. MMM) classes. We also give an application to bundles with fiber …

Effective divisors in the projectivized Hodge bundle

We compute the class of the closure of the locus of canonical divisors in the projectivization of the Hodge bundle $\mathbb{P}\overline{\mathcal{H}}_g$ over $\overline{\mathcal{M}}_g$ which have a zero at a Weierstrass point. We also show that the strata of canonical and bicanonical divisors with a double zero span extremal rays of the respective pseudoeffective cones.

环面上球面丛间的等变映射与KO度

Equivariant maps between sphere bundles over tori and KO-degree

本文给出了环面上球面丛之间的一类等变映射的不存在性结果。证明中使用了等变KO度的概念。作为Seiberg-Witten理论的一个应用，对于具有不定交形式的连通闭定向自旋4-流形，我们得到了第二贝蒂数在第一上同调群和由Albanese映射导出的KO-理论中的某些映射上的一个新的由签名和由四重上积确定的非负整数确定的界.

A counting invariant for maps into spheres and for zero loci of sections of vector bundles

The set of unrestricted homotopy classes $[M,S^n]$ where $M$ is a closed and connected spin $(n+1)$-manifold is called the $n$-th cohomotopy group $π^n(M)$ of $M$. Moreover it is known that $π^n(M) = H^n(M;\mathbb Z) \oplus \mathbb Z_2$ by methods from homotopy theory. We will provide a geometrical description of the $\mathbb Z_2$ part in $π^n(M)$ analogous to Pontryagin's computation of the stable homotopy group $π_{n+1}(S^n)$. This $\mathbb Z_2$ number can be computed by counting embedded circles in $M$ with a certain framing of their normal bundle. This is a analogous result to the mod $2$ …

Characteristic numbers of manifold bundles over surfaces with highly connected fibers

We study smooth bundles over surfaces with highly connected almost parallelizable fiber $M$ of even dimension, providing necessary conditions for a manifold to be bordant to the total space of such a bundle and showing that, in most cases, these conditions are also sufficient. Using this, we determine the characteristic numbers realized by total spaces of bundles of this type, deduce divisibility constraints on their signatures and $\hat{A}$-genera, and compute the second integral cohomology of ${\rm BDiff}^+(M)$ up to torsion in terms of generalized Miller--Morita--Mumford classes. We also pr…

Signature, Toledo invariant and surface group representations in the real symplectic group

In this paper, by using Atiyah-Patodi-Singer index theorem, we obtain a formula for the signature of a flat symplectic vector bundle over a surface with boundary, which is related to the Toledo invariant of a surface group representation in the real symplectic group and the Rho invariant on the boundary. As an application, we obtain a Milnor-Wood type inequality for the signature. In particular, we give a new proof of the Milnor-Wood inequality for the Toledo invariant in the case of closed surfaces and obtain some modified inequalities for the surface with boundary.