Paul Seidel
These are notes from the 2003 C.I.M.E. summer school "symplectic 4-manifolds and algebraic surfaces". They cover the same material as the author's (by now ancient) Ph.D. thesis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We use quilted Floer theory to generalize Seidel's long exact sequence in symplectic Floer theory to fibered Dehn twists. We then apply it to construct versions of the Floer and Khovanov-Rozansky exact triangles in Lagrangian Floer theory of moduli spaces of bundles.
We establish an infinitesimal version of fragility for squared Dehn twists around even dimensional Lagrangian spheres. The precise formulation involves twisting the Fukaya category by a closed two-form or bulk deforming it by a half-dimensional cycle. As our main application, we compute the twisted and bulk deformed symplectic cohomology of the subflexible Weinstein manifolds constructed in \cite{murphysiegel}.
Let $(M^4,ω)$ be a geometrically bounded symplectic manifold, and $L\subset M$ a Lagrangian nodal sphere such that $ω\mid_{π_2(M,L)}=0$. We show that an equatorial Dehn twist of $L$ does not extend to a Hamiltonian diffeomorphism of $M$. We also confirm a mirror symmetry prediction about the action of a symplectomorphism extending an equatorial Dehn twist on the Floer theory of the nodal sphere. We present analogues of the equatorial Dehn twist for more singular Lagrangians, and make concrete conjectures about them.
We study the Floer cohomology of the Dehn twist along a real Lagrangian sphere in a symplectic manifold endowed with an anti-symplectic involution. We prove that there exists a distinguished element in the Floer group that is a fixed point of the automorphism induced by the involution. Our methods of proof are based on Mak-Wu's cobordism and Floer-theoretic considerations.
A long exact sequence on the composition of Dehn twists
本文证明了关联一组 Lagrangian 球的 Floer 同调与其迭代 Dehn twist 不动点 Floer 同调的正合序列，推进了 Seidel 猜想。
In this paper the 5-dimensional contact SO(3)-manifolds are classified up to equivariant contactomorphisms. The construction of such manifolds with singular orbits requires the use of generalized Dehn twists. We show as an application that all simply connected 5-manifoldswith singular orbits are realized by a Brieskorn manifold with exponents (k,2,2,2). The standard contact structure on such a manifold gives right-handed Dehn twists, and a second contact structure defined in the article gives left-handed twists.
Two commuting symplectomorphisms of a symplectic manifold give rise to actions on Floer cohomologies of each other. We prove the elliptic relation saying that the supertraces of these two actions are equal. In the case when a symplectomorphism $f$ commutes with a symplectic involution, the elliptic relation provides a lower bound on the dimension of $HF^*(f)$ in terms of the Lefschetz number of $f$ restricted to the fixed locus of the involution. We apply this bound to prove that Dehn twists around vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have infinite order in t…
Fixed point Floer cohomology of disjoint Dehn twists on a w+-monotone manifold with rational symplectic form
本文显式描述了 w+-单调有理辛流形中不相交 Lagrangian 球面 Dehn 扭曲的 Floer 上同调，通过计数半带条在单调辛流形中定义并证明了一类不动点 Floer 上同调的消失性，计划借此为 Seidel 长正合序列提供新的几何证明。
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.
We initiate the study of the $C^0$ symplectic mapping class group, i.e. the group of isotopy classes of symplectic homeomorphisms. We prove that none of the different powers of the square of the Dehn-Seidel twist belong to the same connected component of the group of symplectic homeomorphisms of certain Liouville domains. This generalizes to the $C^0$ setting a celebrated result of Seidel. In other words, we obtain the non-triviality of the $C^0$ symplectic mapping class group in these domains and in fact an element of infinite order. For that purpose, we develop a method coming from Floer the…
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.
Product and Coproduct on Fixed Point Floer Homology of Positive Dehn Twists
本文计算了单个Dehn扭转迭代不动点Floer同调的积与余积结构，证明其由扭转区域补空间的Morse同调及$T^*S^1$辛同调的特定扇区决定，并通过$J-$全纯截面枚举完成推导。
We define a twisted version of Manolescu and Woodward's Symplectic Instanton homology, prove that this invariant fits into the framework of Wehrheim and Woodward's Floer Field theory, and describe its behaviour for connected sum and Dehn surgery.
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.
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.
Let Y be a compact, oriented 3-manifold with a contact form a. For any Dirac operator D, we study the asymptotic behavior of the spectral flow between D and D+cl(-ira) as r very large. If a is the Thurston-Winkelnkemper contact form whose monodromy is the product of Dehn twists along disjoint circles, we prove that the next order term of the spectral flow function is of order r.
Consider cotangent bundles of exotic spheres, with their canonical symplectic structure. They admit automorphisms which preserve the part at infinity of one fibre, and which are analogous to the square of a Dehn twist. Pursuing that analogy, we show that they have infinite order up to isotopy (inside the group of all automorphisms with the same behaviour).
In this paper we introduce the following new ingredients: (1) rework on part of the Lagrangian surgery theory; (2) constructions of Lagrangian cobordisms on product symplectic manifolds; (3) extending Biran-Cornea Lagrangian cobordism theory to the immersed category. As a result, we manifest Seidel's exact sequences (both the Lagrangian version and the symplectomorphism version), as well as Wehrheim-Woodward's family Dehn twist sequence (including the codimension-1 case missing in the literature) as consequences of our surgery/cobordism constructions. Moreover, we obtain an expression of the a…
The well-known fact that any genus $g$ symplectic Lefschetz fibration $ X^{4}\to S^{2}$ is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manifolds that are realized as symplectic Lefschetz fibrations. In this article we provide new words in the mapping class group, hence new symplectic Lefschetz fibrations. We also compute the signatures of those symplectic Lefschetz fibrations.
We introduce an essential open book foliation, a refinement of the open book foliation, and develop technical estimates of the fractional Dehn twist coefficient (FDTC) of monodromies and the FDTC for closed braids, which we introduce as well. As applications, we quantitatively study the `gap' of overtwisted contact structures and a non-right-veering monodromies. We give sufficient conditions for a 3-manifold to be irreducible and atoroidal. We also show that the geometries of a 3-manifold and the complement of a closed braid are determined by the Nielsen-Thurston types of the monodromies of th…
The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain embedded pseudoholomorphic curves in R cross the mapping torus. It is conjectured to recover the Seiberg-Witten Floer homology of the mapping torus for most spin-c structures, and is related to a variant of contact homology. In this paper we compute the periodic Floer homology of some Dehn twists.
The complex manifold CP^n x CP^{n+1} with symplectic form σ_μ=σ_{CP^n}+μσ_{CP^{n+1}}, where σ_{CP^n} and σ_{CP^{n+1}} are normalized Fubini-Study forms, n a natural number and μ>1 a real number, contains a natural Lagrangian sphere L^μ. We prove that the Dehn twist along L^μ is symplectically isotopic to the identity for all μ>1. This isotopy can be chosen so that it pointwise fixes a complex hypersurface in CP^n x CP^{n+1} and lifts to the blow-up of CP^n x CP^{n+1} along a complex n-dimensional submanifold.
We derive a new exact sequence in the hat-version of Heegaard Floer homology. As a consequence we see a functorial connection between the invariant of Legendrian knots and the contact element. As an application we derive two vanishing results of the contact element making it possible to easily read off its vanishing out of a surgery presentation in suitable situations.
Fixed Point Floer Cohomology of Dehn Twists I: Splitting Formulas
该论文为系列首篇，研究德恩扭转的不动点弗洛尔上同调。利用新限制结果，开发高维情形下计算工具，证明其乘积和微分在余链层面可分解为局部和莫尔斯理论贡献，还提及对未来工作的应用与探讨。
For an open book decomposition $(S,ϕ)$, the fractional Dehn twist coefficients are rational numbers measuring the amount that the monodromy $ϕ$ twists the surface $S$ near each boundary component. In general, the twist coefficients do not behave nicely under the operation of capping off a boundary component. The goal of this paper is to use Heegaard Floer homology to constrain the behavior of the fractional Dehn twist coefficients after capping off. We also use our results about fractional Dehn twists to study the Floer homology of cyclic branched covers over fibered two-component links.
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。
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…
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。
Dynamics of composite symplectic Dehn twists
本文研究复合辛Dehn扭转的非均匀双曲性，证明其具有正拓扑熵和局部稳定/不稳定拉格朗日流形，并给出A²_m构型的辛映射类群分类。
A four-dimensional mapping class group relation
本研究利用全纯曲线技术，建立了四维Weinstein域辛映射类群中右手Dehn扭转乘积间的关系，关键依赖del Pezzo曲面辛子流形辛同痕问题的解，并在附录给出纤维化Dehn扭转与Dehn扭转乘积关系的替代证明。
We prove the existence of a spectral sequence for Lagrangian Floer homology which converges to the Floer homology of the image of a Lagrangian submanifold under multiple fibred Dehn twists. The $E_1$ term of the sequence is given by the hypercube of "resolutions" of the Dehn twists involved. The proof relies on the exact triangle for fibered Dehn twists due to Wehrheim and Woodward. As applications we obtain a spectral sequences from Khovanov homology to symplectic Khovanov homology. Also when a 3-manifold $M$ is given by gluing two handlebodies by a surface diffeomorphism $ϕ$, we obtain a spe…
A new proof is given that Seidel's generalized Dehn twist is not symplectically isotopic to the idenity.
We study Dehn twists along Lagrangian submanifolds that are finite quotients of spheres. We decribe the induced auto-equivalences to the derived Fukaya category and explain its relation to twists along spherical functors.
These notes were prepared in occasion of a mini-course given by the author at the "CIMPA Research School - Hamiltonian and Lagrangian Dynamics" (10-19 March 2015 - Salto, Uruguay). The talks were meant as an introduction to the problem of finding periodic orbits of prescribed energy for autonomous Tonelli Lagrangian systems on the twisted cotangent bundle of a closed manifold. In the first part of the lecture notes, we put together in a general theorem old and new results on the subject. In the second part, we focus on an important class of examples: magnetic flows on surfaces. For such system…
There exists a (relatively minimal) genus g Lefschetz fibration with only one singular fiber over a closed (Riemann) surface of genus h iff g>2 and h>1. The singular fiber can be chosen to be reducible or irreducible. Other results are that every Dehn twist on a closed surface of genus at least three is a product of two commutators and no Dehn twist on any closed surface is equal to a single commutator.
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…
We establish constraints on the topology of smooth Lefschetz fibrations with $4$-dimensional fibers, by studying the family Bauer-Furuta invariant. To compute this invariant, we analyze the framed bordism class of 1-dimensional Seiberg-Witten moduli spaces using the local index theorem by Bismut-Freed. Using this, we deduce new obstructions to the smooth isotopy to the identity for compositions of Dehn twists on $(-2)$-spheres in closed $4$-manifolds. We obtain several applications: (1) We exhibit the first examples of closed simply-connected symplectic $4$-manifolds admitting Torelli symplect…
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…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。