Naoyuki Monden, Kazuya Yoshihara
We give new upper bounds on the stable commutator lengths of Dehn twists along separating curves in the mapping class group of a closed oriented surface. The estimates of these upper bounds are $O(1/g)$, where $g$ is the genus of the surface.
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Stable commutator length of a Dehn twist
证明了映射类群中Dehn扭的稳定换位子长度为正，且关于非分离简单闭曲线的Dehn扭的10次方是两个换位子的乘积.作为应用，给出了德恩捻增长率是线性增长的新证明。
The level $2$ mapping class group of an orientable closed surface can be generated by squares of Dehn twists about non-separating curves. On the other hand, the level $2$ mapping class group $\mathcal{M}_2(N_g)$ of a non-orientable closed surface $N_g$ can not be generated by only Dehn twists, and so it can not be generated by squares of Dehn twists about non-separating curves. In this paper, we prove that the Dehn twist subgroup of $\mathcal{M}_2(N_g)$ can not be generated by squares of Dehn twists about non-separating curves either. As an application, we give a finite generating set for the …
We give new upper bounds on the stable commutator lengths of Dehn twists in mapping class groups and new lower bounds on the stable commutator lengths of Dehn twists in hyperelliptic mapping class groups. In particular, we show that the stable commutator lengths of Dehn twists about a nonseparating and a separating curve on an oriented closed surface of genus 2 are not equal to each other.
We show that on a nonorientable surface of genus at least 7 any power of a Dehn twist is equal to a single commutator in the mapping class group and the same is true, under additional assumptions, for the twist subgroup, and also for the extended mapping class group of an orientable surface of genus at least 3.
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…
Boundary Dehn twists are often commutators
对于偶复维数的完全交或它们的连通和（或更一般地，某些广泛类别的光滑流形），具体构造了带边穿孔X的微分同胚a,c，其交换子[a,c]表示边界Dehn扭转的光滑映射类（相对边界），表明在Baraglia-Konno、Kronheimer-Mrowka、J. Lin和Tilton的工作中已知在相对边界的光滑映射类群中非平凡的4流形上的边界Dehn扭转在交换化后变为平凡，推广了Y. Lin的结果。
When can surgery on a null-homologous knot K in a rational homology sphere produce a non-separating sphere? We use Heegaard Floer homology to give sufficient conditions for K to be unknotted. We also discuss some applications to homology cobordism, concordance, and Mazur manifolds.
We give a new upper bound on the stable commutator length of Dehn twists in hyperelliptic mapping class groups, and determine the stable commutator length of some elements. We also calculate values and the defects of homogeneous quasimorphisms derived from ω-signatures, and show that they are linearly independent in the mapping class groups of pointed 2-spheres when the number of points is small.
Let $N_g^k$ be a nonorientable surface of genus \ $g\geq 5$ \ with \ $k$-punctures. In this note, we will give an algebraic characterization of a Dehn twist about a simple closed curve on $N_g^k$. Along the way, we will fill some little gaps in the proofs of some theorems in \cite{A} and \cite{I1} giving algebraic characterizations of Dehn twists about separating simple closed curves. Indeed, our results will give an algebraic characterization for the topological type of Dehn twists about separating simple closed curves.
Let $C$ be a curve in a closed orientable surface $F$ of genus $g \geq 2$ that separates $F$ into subsurfaces $\widetilde {F_i}$ of genera $g_i$, for $i = 1,2$. We study the set of roots in $\Mod(F)$ of the Dehn twist $t_C$ about $C$. All roots arise from pairs of $C_{n_i}$-actions on the $\widetilde{F_i}$, where $n=\lcm(n_1,n_2)$ is the degree of the root, that satisfy a certain compatibility condition. The $C_{n_i}$ actions are of a kind that we call nestled actions, and we classify them using tuples that we call data sets. The compatibility condition can be expressed by a simple formula, al…
Profinite isomorphisms, stable commutator length, and fixed point properties
本文通过构造Grothendieck对，证明稳定交换子长度、准同态等不是有限群不变量，并结合Rips构造与Dehn填充方法进行研究。
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。
The Magnus representation of the Torelli subgroup of the mapping class group of a surface is a homomorphism r: I_{g,1} -> GL_{2g}(Z[H]). Here H is the first homology group of the surface. This representation is not faithful; in particular, Suzuki previously described precisely when the commutator of two Dehn twists about separating curves is in the kernel of r. Using the trace of the Magnus representation, we apply a new method of showing that two endomorphisms generate a free group to prove that the images of two positive separating multitwists under the Magnus representation either commute o…
We show that if a hyperbolic knot manifold $M$ contains an essential twice-punctured torus $F$ with boundary slope $β$ and admits a filling with slope $α$ producing a Seifert fibred space, then the distance between the slopes $α$ and $β$ is less than or equal to $5$ unless $M$ is the exterior of the figure eight knot. The result is sharp; the bound of $5$ can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the $α$-filling contains no non-abelian free group. The proofs are divided into the four cases $F$ is a …
The Dehn Twist Action for Quantum Representations of Mapping Class Groups
本文研究了量子映射类群中德尼特旋作用的性质，通过分析非分离和分离情况下的顺序，推广了相关结果并探讨了约翰森核与Torelli群的平凡作用。
Finiteness of canonical quotients of Dehn quandles of surfaces
本文研究闭可定向曲面Dehn广群的典范商有限性，刻画其2-广群，确定n-广群有限的n值，计算最小非平凡商大小，还证明Artin广群对合商为Coxeter广群并确定辫子广群最小非平凡商。
We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove that the level 2 twist subgroup is normally generated in the mapping class group by a crosscap pushing map along a non-separating two-sided simple loop for genus $g\geq 5$ and $g=3$. As an application, we calculate the first homology group of the level 2 twist subgroup for genus $…
A generating set for the Johnson kernel
本文证明双曲曲面的 Johnson 核可由界定特定低亏格子曲面或带点圆盘的分离简单闭曲线上的 Dehn 扭转生成，给出了该核的一个更精细生成集。
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.…
Let $Σ$ be a compact oriented surface. The Dehn twist along every simple closed curve $γ\subset Σ$ induces an automorphism of the fundamental group $π$ of $Σ$. There are two possible ways to generalize such automorphisms if the curve $γ$ is allowed to have self-intersections. One way is to consider the `generalized Dehn twist' along $γ$: an automorphism of the Malcev completion of $π$ whose definition involves intersection operations and only depends on the homotopy class $[γ]\in π$ of $γ$. Another way is to choose in the usual cylinder $U:=Σ\times [-1,+1]$ a knot $L$ projecting onto $γ$, to p…
The purpose of this article is two-fold: We first give a more elementary proof of a recent theorem of Korkmaz, Monden, and the author, which states that the commutator length of the n-th power of a Dehn twist along a boundary parallel curve on a surface with boundary S of genus g at least two is the floor of (|n|+3)/2 in the mapping class group of S. The alternative proof we provide goes through push maps and Morita's use of Milnor-Wood inequalities, in particular it does not appeal to gauge theory. In turn, we produce infinite families of pairwise non-homotopic 4-manifolds admitting genus g s…
Exotic Dehn twists on sums of two contact 3-manifolds
本文利用推广的 Kronheimer–Mrowka 接触不变量与凸球面 h-原理，证明两个闭接触3-流形连通和上的 Dehn 扭转可形成无限阶奇异接触映射类，并给出 Colin 分解定理的参数化版本及过扭转情形的新现象。
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.
By Torelli topology the author understands aspects of the topology of surfaces (potentially) relevant to the study of Torelli groups. The present paper is devoted to a new approach to the results of W. Vautaw about Dehn multi-twists in Torelli groups and abelian subgroups of Torelli groups. The new proofs are more transparent and lead to stronger estimates of the rank of an abelian subgroup when some additional information is available. An unexpected application of our methods is a new proof of the algebraic characterization of the Dehn twist about separating circles and the Dehn-Johnson twist…
Exotic Dehn twists on 4-manifolds
该研究率先探索4-流形内沿非$S^3$ 3-流形的奇异Dehn扭，借助$\text{RP}^2$上的2参数族Seiberg-Witten理论，构造出多类新的奇异微分同胚例子，还证明了相关微分同胚群基本群同态的非满射性。
We show that, for certain families $ϕ_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $ϕ^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We sho…
Handle additions producing essential surfaces
我们构造了一个小的双曲3-流形$M$，使得对于任意整数$g\geq 2$，在$\partial M$中有无穷多个分离斜率$r$，使得沿$r$给$M$附加一个2-柄得到的3-流形$M(r)$是双曲的，并且包含$g$的本质分离闭曲面.该结果与已知的Dehn填充的有限性结果形成了鲜明的对比，也与已知的$g=0,1$的柄加法的有限性结果形成了鲜明的对比.我们的3-流形$M$是亏格3的双曲小纽结的补。
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.
Perturbed Floer Homology of some Fibered Three Manifolds II
本文改进了文献[21]的方法，计算了几类特殊的纤维三流形在次高自旋$^c$ -结构$S_{g-2}$中的扰动$HF^+$.本文所考虑的特殊类包括沿一条非分离曲线和沿一对横截曲线的Dehn扭曲的映射环面。
We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book --- except possibly for a few low genera cases. To obtain our results, we produce the first examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary. This solves an open problem posed by…
On the kernel of $\mathrm{SO}(3)$-Witten-Reshetikhin-Turaev quantum representations
本文研究闭曲面映射类群的SO(3)-Witten-Reshetikhin-Turaev量子表示的核，证明在亏格和素数满足一定条件时，该核包含于由Dehn扭转的p次幂、分离扭转或Johnson子群换位子群生成的子群中。
We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.
Cutting and Pasting in the Torelli Group
我们引入机器，以允许"剪切和粘贴"式的归纳参数的Torelli子群的映射类群。在过去，这些论点一直是有问题的，因为限制Torelli组的子表面给不同的群体，这取决于子表面是如何嵌入的。我们定义了一个类别$\TSur$，其对象是表面连同装饰限制他们如何可以嵌入到更大的表面和其态射的嵌入尊重装饰。在这个范畴上有一个自然的"Torelli函子"，它扩展了Torelli群在闭曲面上的通常定义。此外，我们证明了一个类似的Birman精确序列的Torelli组的表面与边界和使用的行动Torelli组的复杂的曲线找到发电机的Torelli组。对于亏格$g \geq 1$，只需要围绕（某些）分离曲线和边界对的扭转，而对于亏格$g=0$，需要一种新型的生成元（"简单相交对的交换子"）。作为一个特殊的情况下，我们的方法提供了一个新的，更概念性的证明的经典结果的Birman-Powell说，Torelli组的闭曲面上产生的扭曲分离曲线和边界对。
Stabilizing Heegaard splittings of toroidal 3-manifolds
设$T$是三维流形$M$中的不可压缩分离环面。假设一个亏格$g$ Heegaard分裂$V \cup_S W$可以相对于$T$很好地定位（例如，$V \cup_S W$是强不可约的），我们得到了一个上界的稳定化所需的$V \cup_S W$成为同位素的一个Heegaard分裂，这是一个沿$T$的合并。特别地，如果$T$是$M$的JSJ分解中的规范环面，则必要的稳定化的数量至多为$4g-4$。作为推论，这建立了$V \cup_S W$和任何Heegaard分裂所需的稳定化数目的上界，所述Heegaard分裂是通过$V \cup_S W$沿$T$的Dehn扭曲获得的，以成为同位素。
Heegaard Floer homology of certain mapping tori
我们计算了映射环面M与某些曲面超同态相关联的Heegaard弗洛尔同调$HF^+（M，s），其中s是M上的任何Spin^c结构，其第一陈类是非挠的。设Gamma和Delta是亏格为g的黎曼曲面Sigma_g上的一对几何对偶非分离曲线，Sigma是将Sigma_g分离为亏格1和g-1的分支的曲线.写出关于这些曲线的右手德恩扭曲的t-gamma、t_delta和t_sigma。我们考虑的例子是Z中m，n的双同态t_gamma^m circ t_delta^n的映射环面和t_sigma^{+-1}的映射环面。
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