Peter Kronheimer, Tomasz Mrowka
Ruberman gave the first examples of self-diffeomorphisms of four-manifolds that are isotopic to the identity in the topological category but not smoothly so. We give another example of this phenomenon, using the Dehn twist along a 3-sphere in the connected sum of two K3 surfaces.
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We compute the Reidemeister torsion of the complement of a twist knot in $S^3$ and that of the 3-manifold obtained by a Dehn surgery on a twist knot.
Dehn twists and the Nielsen realization problem for spin 4-manifolds
本研究针对非零符号差的闭定向光滑自旋4维流形，证明围绕±2-球面的Dehn扭转不与有限阶微分同胚同伦，否定了相关映射类群的Nielsen实现问题，还发现K3与S²×S²连通和的拓扑与光滑范畴Nielsen实现问题存在差异，核心工具是Kato的对合10/8型不等式及其改进。
We give Dehn twist--crosscap slide presentations for involutions on non-orientable surfaces of genera up to 5.
We classify groups generated by powers of 2 Dehn twists which are 1) free or 2) have no ``unexpected'' reducible elements. We give some sufficient conditions in the case of groups generated by powers of more than two twists.
Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g \geq 1$. In this paper, we develop various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy. As applications, we develop methods for factoring certain roots of Dehn twists as words in Dehn twists. We will also show the existence of conjugates of periodic maps of order $4g$ and $4g+2$, for $g\geq 2$, whose product is pseudo-Anosov.
We give a parity condition of a Heegaard diagram to show that it is unstabilized. This improves the result of [5]. As an application, we construct unstabilized Heegaard splittings by Dehn twists on any given Heegaard splitting.
We construct nontrivial roots of Dehn twists about nonseparating curves.
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.
A positive Dehn twist product for a $\mathbb{Z}_3$ action with $g+2$ fixed points on the 2-dimensional closed, compact, oriented surface $Σ_g$ is presented. The homeomorphism invariants of the resulting symplectic 4-manifolds are computed.
We establish a relationship between Heegaard Floer homology and the fractional Dehn twist coefficient of surface automorphisms. Specifically, we show that the rank of the Heegaard Floer homology of a 3-manifold bounds the absolute value of the fractional Dehn twist coefficient of the monodromy of any of its open book decompositions with connected binding. We prove this by showing that the rank of Floer homology gives bounds for the number of boundary parallel right or left Dehn twists necessary to add to a surface automorphism to guarantee that the associated contact manifold is tight or overt…
In this note, we establish a relationship between fractional Dehn twist coefficients of Riemann surface automorphisms and modular invariants of holomorphic families of algebraic curves. Specially, we give a characterization of pseudo-periodic maps with nontrivial fractional Dehn twist coefficients. We also obtain some uniform lower bounds of non-zero fractional Dehn twist coefficients.
We give an explicit formula for the action of the Dehn twist along a simple closed curve in a compact connected oriented surface on the completion of the filtered skein modules. To do this, we introduce filtrations of the Kauffman bracket skein algebra and the Kauffman bracket skein modules on the surface.
We give a small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists. The difference between the number of the generators and a lower bound of numbers of generators for the twist subgroup by Dehn twists is one. The lower bounds is obtained from an argument of Hirose [5].
We introduce a Lie algebra associated with a non-orientable surface, which is an analogue for the Goldman Lie algebra of an oriented surface. As an application, we deduce an explicit formula of the Dehn twist along an annulus simple closed curve on the surface as in Kawazumi-Kuno and Masseyeau-Turaev.
A \textit{multicurve} $\C$ on a closed orientable surface is defined to be a finite collection of disjoint non-isotopic essential simple closed curves. The Dehn twist $t_{\C}$ about $\C$ is the product of the Dehn twists about the individual curves. In this paper, we give necessary and sufficient conditions for the existence of a root of such a Dehn twist, that is, a homeomorphism $h$ such that $h^n = t_{\C}$. We give combinatorial data that corresponds to such roots, and use it to determine upper bounds for $n$. Finally, we classify all such roots up to conjugacy for surfaces of genus 3 and 4…
Given a 3-holed sphere decomposition of an orientable closed surface, it is shown that each orientation preserving homeomorphism of the surface is isotopic to a composition AB where A is a product of positive Dehn twists and B is a product of negative Dehn twists on the decomposition curves.
This note records the order of a higher dimensional Dehn twist in a range of topologically significant groups.
The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-intersection, it is induced from the usual Dehn twist along the curve. In this expository article, after explaining their definition, we review several results about generalized Dehn twists such as their realizability as diffeomorphisms of the surface, their diagrammatic description …
Positive Dehn twist products for some elements of finite order in the mapping class group of a 2-dimensional closed, compact, oriented surface $Σ_g$, which are rotations of $Σ_g$ through $2π/p$, are presented. The homeomorphism invariants of the resulting simply connected symplectic 4- manifolds are computed.
Let f be a Z/2Z-spin structureon a closed surface S of genus g>3. We determine a generating set of the stabilizer of f in the mapping class group of S consisting of Dehn twists about an explicit collection of 2g+1 curves in S. If g=3 then we also determine a generating set of the stabilizer of an odd Z/2Z-spin structure consisting of Dehn twists about a collection of 6 curves.
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 …
Knot Floer homology of boundary Dehn twists
本文证明纽结Floer复形及翻转映射限制可确定Borromean纽结$\pm 1$手术的对偶纽结，进而表明Heegaard Floer同调能检测边界Dehn扭转。
Fractional Dehn twist coefficients and rank bounds for categorified link invariants
该研究给出分层纽结不变量秩的两个新下界，分别对应纤维纽结的纽结弗洛尔同调和辫子闭包的环形霍万诺夫同调，核心技术是确定纤维纽结单值群添加边界Dehn扭转时的同调行为。
Let $N_g$ be the non-orientable surface with genus $g$, $\text{MCG}(N_g)$ be the mapping class group of $N_g$, $\mathcal{T}(N_g)$ be the index 2 subgroup generated by all Dehn twists of $\text{MCG}(N_g)$. We prove that for odd genus, $\mathcal{T}(N_g)$ can be generated by three elements of finite orders.
Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Futhermore, we give data set which determine the conjugacy class of a root. As a consequence, we can find the minimum degree and the maximum degree, and show that the degree must be odd. Also, we give Dehn twist expression of the root of degree 3.
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.
In this note we give presentations of all finite subgroups of the mapping class group of a closed surface of genus 2 by the Humphries generators up to conjugacy.
We discuss how the fractional Dehn twist coefficient behaves under a fully ramified branched covering of an open book, and give applications to both topological and contact 3-manifolds. Among them, we show that non-right-veering closed braids represent virtually loose transverse links.
The positive Dehn twist expressions for the generalizations of the new involutions described in math.GT/0404310 are presented. The homeomorphism types of the Lefschetz fibrations that they define are determined for several examples.
It is proved that the stable commutator length of a Dehn twist in the mapping class group is positive and the tenth power of a Dehn twist about a nonseparating simple closed curve is a product of two commutators. As an application a new proof of the fact that the growth rate of a Dehn twist is linear is given.
Let a and b be two simple closed curves on an orientable surface S such that their geometric intersection number is greater than 1. It is known that the group generated by corresponding Dehn twists t_a and t_b is isomorphic to the free group of rank 2. In this paper we extend this result to the case of a nonorientable surface.
We study subgroups of the mapping class group of the torus generated by powers generated by powers of Dehn twists. We give a criterion to show when a collection of powers Dehn twists generates a free group using the ping pong lemma. We show that the subgroup generated by three uniform powers of Dehn twists can be either the whole mapping class group, a direct product of a free group of rank two with the cyclic group of order two, or a free group of rank at most three. We characterize subgroups generated by a collection of (respectively squares of) Dehn twists, if there is a pair among so that …
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.
A note on the boundary Dehn twist of $K3$ surfaces
本文针对带孔K3曲面的边界Dehn扭转，证明其在光滑映射类群阿贝尔化后平凡，证明依托Baraglia-Konno的Spin^C族障碍与K3曲面整体Torelli定理。
Exotic Dehn twists and homotopy coherent group actions
本文研究将光滑4-流形边界上的同伦相干有限循环群作用延拓至内部的问题，并证明沿任意Seifert同调球面（3-球面除外）在单连通正定填充上的Dehn扭转具有无限阶奇异性。
Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichmüller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the Teichmüller space. In contrast we show the number of Dehn twist lattice points intersecting a closed ball of radius $R$ is coarsely asymptotic to $e^{\frac{h}{2}R}$. Moreover, we show the number multi-twist lattice points intersecting a closed ball of radius $R$ grows coarsely at least at the rate of $R \cdot e^{\frac{h}{2}R}$.
A homeomorphism of a 3-manifold M is said to be Dehn twists on the boundary when its restriction to the boundary of M is isotopic to the identity on the complement of a collection of disjoint simple closed curves in the boundary of M. In this paper, we give various results about such collections of curves and the associated homeomorphisms. In particular, if M is compact, orientable, irreducible and the boundary of M is a single torus, and M admits a homeomorphism which is a nontrivial Dehn twist on the boundary of M, then M must be a solid torus.
By using a notion of a geometric Dehn twist in $\sharp_k(S^2 \times S^1)$, we prove that when projections of two $\mathbb{Z}$-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $\mathbb{Z}$-splittings generate a free group of rank $2$. Moreover, every element from this free group is either conjugate to a power of one of the Dehn twists or it is a fully irreducible outer automorphism of the free group. We also prove that, when projected to the intersection graph, the same group of Dehn twists produce ato…
We classify up to conjugacy the group generated by a commuting pair of a periodic diffeomorphism and a hyperelliptic involution on an oriented closed surface. This result can be viewed as a refinement of Ishizaka's result on classification of the mapping classes of hyperelliptic periodic diffeomorphisms. As an application, we obtain the Dehn twist presentations of hyperelliptic periodic mapping classes, which are closely related to the ones obtained by Ishizaka.
D. Margalit and S. Schleimer found examples of roots of the Dehn twist about a nonseparating curve in a closed orientable surface, that is, homeomorphisms whose nth power is isotopic to the Dehn twist. Our main theorem gives elementary number-theoretic conditions that describe the values of n for which an nth root exists, given the genus of the surface. Among its applications, we show that n must be odd, that the Margalit-Schleimer roots achieve the maximum value of n among the roots for a given genus, and that for a given odd n, nth roots exist for all genera greater than (n-2)(n-1)/2. We als…
On four-dimensional Dehn twists and Milnor fibrations
该研究借助磁单极Floer同调进展计算族Seiberg--Witten不变量，证明极小椭圆奇点链不定辛填充的边界Dehn扭转无限阶非平凡，揭示了四维空间中三类全新的拓扑与辛几何现象。
We introduce a notion of a Fox pairing in a group algebra and use Fox pairings to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces. This work is inspired by the Kawazumi-Kuno generalization of the Dehn twists to non-simple closed curves on surfaces.
We examine open books with powers of fibered Dehn twists as monodromy. The resulting contact manifolds can be thought of as Boothby-Wang orbibundles over symplectic orbifolds. Using the mean Euler characteristic of equivariant symplectic homology we can distinguish these contact manifolds and hence show that some fibered Dehn twists are not symplectically isotopic to the identity relative to the boundary. This complements results of Biran and Giroux.
We show that any exceptional non-trivial Dehn surgery on a twist knot, except the trefoil, yields a 3-manifold whose fundamental group is left-orderable. This is a generalization of a result of Clay, Lidman and Watson, and also gives a new supporting evidence for a conjecture of Boyer, Gordon and Watson.
For any hyperbolic twist knot in the 3-sphere, we show that the resulting manifold by $r$-surgery on the knot has left-orderable fundamental group if the slope $r$ satisfies the inequality $0\le r \le 4$.
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 any homologically non-trivial Dehn twist of a compact surface F with boundary is the lifting of a half-twist in the braid group B_n, with respect to a suitable branched covering p : F -> B^2. In particular, we allow the surface to have disconnected boundary. As a consequence, any allowable Lefschetz fibration on B^2 is a branched covering of B^2 x B^2.
In 2001, J. Hempel proved the existence of Heegaard splittings of arbitrarily high distance by using a high power of a pseudo-Anosov map as the gluing map between two handlebodies. We show that lower bounds on distance can also be obtained when using a high power of a suitably chosen Dehn twist. In certain cases, we can then determine the exact distance of the resulting splitting. These results can be seen as a natural extension of work by A. Casson and C. Gordon in 1987 regarding strongly irreducible Heegaard splittings.
For any unoriented loop on a compact connected oriented surface with one boundary component, the generalized Dehn twist along the loop is defined as an automorphism of the completed group ring of the fundamental group of the surface. If the loop is simple, this is the usual right handed Dehn twist, in particular realized as a mapping class of the surface. We investigate the case when the loop has a single transverse double point, and show that in this case the generalized Dehn twist is not realized as a mapping class.
In this short note we construct examples of open books for 3-manifolds that show that arbitrarily high twisting of the monodromy of the open book does not guarantee maximality of the Euler characteristic of the pages among the open books supporting the same contact manifold. We find our examples of open books as the double branched covers of families of closed braids studied by Malyutin and Netsvetaev.
Let t_a be the Dehn twist about a circle a on an orientable surface. It is well known that for each circle b and an integer n, I(t_a^n(b),b)=|n|I(a,b)^2, where I(,) is the geometric intersection number. We prove a similar formula for circles on nonorientable surfaces. As a corollary we prove some algebraic properties of twists on nonorientable surfaces. We also prove that if M(N) is the mapping class group of a nonorientable surface N, then up to a finite number of exceptions, the centraliser of the subgroup of M(N) generated by the twists is equal to the centre of M(N) and is generated by twi…
Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact sequence to arbitrary powers of a fixed Dehn twist. We also show that the Milnor fibre of any isolated degenerate hypersurface singularity contains such pairs of spheres.
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。
Modular invariants of families of curves are Arakelov invariants in arithmetic algebraic geometry. All the known uniform lower bounds of these invariants are not sharp. In this paper, we aim to give explicit lower bounds of modular invariants of families of curves, which is sharp for genus 2. According to the relation between fractional Dehn twists and modular invariants, we give the sharp lower bounds of fractional Dehn twist coefficients and classify pseudo-periodic maps with minimal coefficients for genus 2 and 3 firstly. We also obtain a rigidity property for families with minimal modular …
We refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(F_n). This gives restrictions to actions of Aut(F_n) on CAT(0) spaces.
Let N_{g,s} denote the nonorientable surface of genus g with s boundary components. Recently Paris and Szepietowski obtained an explicit finite presentation for the mapping class group M(N_{g,s}) of the surface N_{g,s}, where s\in{0,1} and g+s>3. Following this work we obtain a finite presentation for the mapping class group M(N_{g,s}) with generators being Dehn twists and one crosscap slide.
Hyperbolic Heegaard splittings and Dehn twists
该研究针对满足pared非柱性的曲线粘合的亏格≥2的Heegaard分裂，证明当n≥14时，这类分裂具有双曲度量，对应Dehn扭转的闭曲线为特定长度的闭测地线，丰富了三维流形几何拓扑的相关结论。
Cusp shape and fractional Dehn twists of fibred hyperbolic 3-manifolds
本文将带边纤维化双曲3-流形尖点的欧氏几何与其单值性的分数Dehn扭转系数进行粗略关联，并应用于Dehn填充下的几何估计及接触拓扑中紧接触结构的几何判据。
The Dehn twist on a connected sum of two homology tori
本文推广Pin(2)-等变族Bauer-Furuta不变量至非单连通流形，并构造精细化，证明两个同调环面连通和颈部沿3-球面的Dehn扭转非光滑同伦于恒同，条件为行列式均为奇数。
Boundary Dehn twists on Milnor fibers and Family Bauer--Furuta invariants
本文利用映射环面的 family relative Bauer–Furuta 不变量，证明特定 Brieskorn Milnor 纤维上的边界 Dehn 扭转在给定数论条件下是相对于边界的奇异微分同胚。
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