Tight contact structures on Seifert manifolds over T^2 with one singular fibre

On continuous families of geometric Seifert conemanifold structures

We determine the Thurston's geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space $S^2$ and no more than three exceptional fibres, whose singular set, composed by fibres, has at most 3 components which can include exceptional or general fibres (the total number of exceptional and singular fibres is less or equal than three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery in a torus knot $K_{(r,…

Alexander 多项式、Dijkgraaf-Witten 不变量与 Seifert 纤维化手术

Alexander polynomial, Dijkgraaf-Witten invariant, and Seifert fibred surgery

本文利用 Dijkgraaf-Witten 不变量研究沿纽结的 k/ℓ 手术产生的小 Seifert 3-流形，证明其重数与手术系数受 Alexander 多项式约束。

Chern-Simons Theory with Complex Gauge Group on Seifert Fibred 3-Manifolds

We consider Chern-Simons theory with complex gauge group and present a complete non-perturbative evaluation of the path integral (the partition function and certain expectation values of Wilson loops) on Seifert fibred 3-Manifolds. We use the method of Abelianisation. In certain cases the path integral can be seen to factorize neatly into holomorphic and anti-holomorphic parts. We obtain closed formulae of this factorization for the expectation values of torus knots.

负定Seifert纤维空间上的可填充结构

Fillable structures on negative-definite Seifert fibred spaces

本文研究负定星形连接空间上的可填充接触结构，证明此类Seifert纤维空间具有唯一的负最大扭数，并通过晶格上同调的Alexander滤子计算该扭数，进一步展示这些流形上的负扭紧结构由复表面奇点的最小解析中的Stein结构诱导。

Combinatorial Seifert fibred spaces with transitive cyclic automorphism group

In combinatorial topology we aim to triangulate manifolds such that their topological properties are reflected in the combinatorial structure of their description. Here, we give a combinatorial criterion on when exactly triangulations of 3-manifolds with transitive cyclic symmetry can be generalised to an infinite family of such triangulations with similarly strong combinatorial properties. In particular, we construct triangulations of Seifert fibred spaces with transitive cyclic symmetry where the symmetry preserves the fibres and acts non-trivially on the homology of the spaces. The triangul…

On tight contact structures with negative maximal twisting number on small Seifert manifolds

We study some properties of transverse contact structures on small Seifert manifolds, and we apply them to the classification of tight contact structures on a family of small Seifert manifolds.

Classes d'homotopie de champs de vecteurs Morse-Smale sans singularité sur les fibrés de Seifert

In the first part of this paper, we consider smooth maps from a compact orientable 3-manifold without boundary to the 2-sphere. We give a geometric criterion to decide whether two given maps are homotopic, based on the sets of points where the maps are equal or antipodal. We extend this criterion to non-singular vector fields (and co-oriented plane fields) on 3-manifolds. In the second part, we make use of this criterion to study non-singular Morse-Smale vector fields on Seifert manifolds, giving a simple proof of a result of Yano.

某些 Seifert 纤维化流形上紧接触结构的分类：I

Classification of tight contact structures on some Seifert fibered manifolds: I

本文通过 Legendrian 手术构造接触结构获得下界，并利用凸曲面理论获得上界，对具有四条例外纤维的某些 Seifert 纤维化流形上零 Giroux 挠率的紧接触结构进行了分类。

Minimal triangulation size of Seifert fibered spaces with boundary

One measure of the complexity of a 3-manifold is its triangulation complexity: the minimal number of tetrahedra in a triangulation of it. A natural question is whether we can relate this quantity to its topology. We determine the triangulation complexity of Seifert fibered spaces with non-empty boundary in terms of their Seifert data, up to a multiplicative constant. We also show that all singular fibres of such a Seifert fibered space (aside from those of multiplicity two) can be made simplicial in the 79th barycentric subdivision of any triangulation of it.

The linking pairings of orientable Seifert manifolds

We compute the $p$-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free $S^1$-action. Using this, we show that any non-singular linking pairing on a finite abelian group with homogeneous 2-primary summand is realized by such a manifold. However, some pairings on inhomogeneous 2-groups are not realizable by any Seifert fibred 3-manifold.

实奇异 Del Pezzo 曲面与有理曲线纤维化的三维流形，II

Real singular Del Pezzo surfaces and 3-folds fibred by rational curves, II

本文肯定回答 Kollár 的三个问题，对有理曲面上的实三维流形给出 Seifert 纤维与透镜空间的精确估计，推广 Comessatti 定理，并构造非可定向双曲型基的极小例子。

Tight Contact Small Seifert Spaces with $e_0\neq0,-1,-2$

We classify up to isotopy the tight contact structures on small Seifert spaces with $e_0\neq0,-1,-2$. (The first version contains on the $e_0<-2$ case.)

Classification of tight contact structures on small Seifert fibered L-spaces

The Ozsvath-Szabo contact invariant is a complete classification invariant for tight contact structures on small Seifert fibered 3-manifolds which are L-spaces.

某些小 Seifert 纤维空间族上的紧接触结构

Tight contact structures on some families of small Seifert fiber spaces

本文提出一种通过比较等价 Thurston–Bennequin 标架的 Legendrian 表示来证明手术后接触结构仍不同的方法，并结合凸曲面理论分类若干小 Seifert 纤维空间族上的紧接触结构。

Classification of tight contact structures on small Seifert 3-manifolds with e_0\geq 0

We classify positive, tight contact structures on closed Seifert fibered 3-manifolds with base S^2, three singular fibers and e_0\geq 0.

On the existence of tight contact structures on Seifert fibered 3-manifolds

We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new non-vanishing criterion for the contact Ozsvath-Szabo invariant.

Tight small Seifert fibered manifolds with $e_0=-2$

In this paper we provide the classification of tight contact structures on some small Seifert fibered manifolds. As an application of this classification, combined with work of Lekili in \cite{L2010}, we obtain infinitely many counterexamples to a question of Honda-Kazez-Matić that asks whether a right-veering, non-destabilizable open book necessarily supports a tight contact structure.

Tight contact structures on some small Seifert fibered 3--manifolds

We classify tight contact structures on the small Seifert fibered 3--manifold M(-1; r_1, r_2, r_3) with r_i in (0,1) and r_1, r_2 \geq 1/2. The result is obtained by combining convex surface theory with computations of contact Ozsvath--Szabo invariants. We also show that some of the tight contact structures on the manifolds considered are nonfillable, justifying the use of Heegaard Floer theory.

Compatible contact structures of fibered Seifert links in homology 3-spheres

We study compatible contact structures of fibered Seifert multilinks in homology 3-spheres and especially give a necessary and sufficient condition for the contact structure to be tight in the case where the Seifert fibration is positively twisted. As a corollary we determine the strongly quasipositivity of fibered Seifert links in the 3-sphere. We also study the compatible contact structures of cablings along links in any 3-manifolds.

关于允许共定向紧致叶面但无消失欧拉类的3维流形

On 3-manifolds admitting co-orientable taut foliations, but none with vanishing Euler class

本文构造了允许共定向紧致叶面但无消失欧拉类的3维流形，揭示了其与L空间猜想及左可序基本群的关系。

Tight contact structures on Seifert surface complements

We consider complements of standard Seifert surfaces of special alternating links. On these handlebodies, we use Honda's method to enumerate those tight contact structures whose dividing sets are isotopic to the link, and find their number to be the leading coefficient of the Alexander polynomial. The Euler classes of the contact structures are identified with hypertrees in a certain hypergraph. Using earlier work, this establishes a connection between contact topology and the Homfly polynomial. We also show that the contact invariants of our tight contact structures form a basis for sutured F…

Adiabatic limits of Seifert fibrations, Dedekind sums, and the diffeomorphism type of certain 7-manifolds

We extend the adiabatic limit formula for eta-invariants by Bismut-Cheeger and Dai to Seifert fibrations. Our formula contains a new contribution from the singular fibres that takes the form of a generalised Dedekind sum. As an application, we compute the Eells-Kuiper and t-invariants of certain cohomogeneity one manifolds that were studied by Dearricott, Grove, Verdiani, Wilking, and Ziller. In particular, we determine the diffeomorphism type of a new manifold of positive sectional curvature.

Seifert fibered contact three-manifolds via surgery

Using contact surgery we define families of contact structures on certain Seifert fibered three-manifolds. We prove that all these contact structures are tight using contact Ozsath-Szabo invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three-manifold carrying at least n pairwise non-isomorphic tight, not fillable contact structures.

Heegaard Floer同调与最大扭转数

Heegaard Floer homology and maximal twisting numbers

本文通过调整Ozsváth-Szabó的完整路径算法，建立了Seifert纤维空间上负扭转紧致接触结构与Heegaard Floer同调群之间的对应关系，揭示了这些结构的分类及其接触不变量c^+的区分。

具有单个JSJ环面的SU(2)-阿贝尔图流形

SU(2)-abelian graph manifolds with a single JSJ torus

该研究针对粘合两个带两条奇异纤维的圆盘上Seifert纤维化空间得到的单JSJ环面图流形，依据Seifert系数完成SU(2)-阿贝尔3-流形的分类，并证明这类流形是Heegaard Floer同调L-空间。

Tight contact structures on some bounded Seifert manifolds with minimal convex boundary

We classify positive tight contact structures, up to isotopy fixing the boundary, on the manifolds $N=M(D^{2}; r_1, r_2)$ with minimal convex boundary of slope $s$ and Giroux torsion 0 along $\partial N$, where $r_1,r_2\in (0,1)\cap\mathbb{Q}$, in the following cases: (1) $s\in(-\infty, 0)\cup[2, +\infty)$; (2) $s\in[0, 1)$ and $r_1,r_2\in [1/2,1)$; (3) $s\in[1, 2)$ and $r_1,r_2\in(0,1/2)$; (4) $s=\infty$ and $r_1=r_2=1/2$. We also classify positive tight contact structures, up to isotopy fixing the boundary, on $M(D^2;1/2,1/2)$ with minimal convex boundary of arbitrary slope and Giroux torsio…

Dehn fillings of knot manifolds containing essential twice-punctured tori

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 …

Ozsvath-Szabo invariants and tight contact three-manifolds, III

We characterize L-spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain contact Seifert fibered 3-manifolds with e_0=-1 to have nonzero contact Ozsvath--Szabo invariants. This yields an algorithm for deciding whether a given small Seifert fibered L-space carries a contact structure with nonvanishing contact Ozsvath--Szabo invariant. As an application, we prove the existence of tight contact structures on some 3-manifolds obtained by integral surgery along a positive tor…

On the classification of tight contact structures

Recently, there have been several breakthroughs in the classification of tight contact structures. We give an outline on how to exploit methods developed by Ko Honda and John Etnyre to obtain classification results for specific examples of small Seifert manifolds.

小Seifert L-空间及更多类别的三维辛化问题

The three-dimensional symplectization question for small Seifert L-spaces and further classes

本文通过构造Liouville配边，解决了小Seifert L-空间等若干三维流形的辛化问题，证明了强可填充性等刚性性质。

Intersection Pairings on Spaces of Connections and Chern-Simons Theory on Seifert Manifolds

Let M be a U(1) bundle over a smooth Riemann surface. I show that for Chern-Simons theory on M, with structure group G, the path integral is an integral over the space of G-connections on the Riemann surface involving characteristic classes as well as a certain 4-dimensional class that comes from a universal bundle. When M is the product of a Riemann surface with a circle the 4-dimensional class does not enter and the path integral takes the form of a Riemann-Roch formula albeit in infinite dimensions. The discussion is generalised to include Wilson lines along the fibre direction in M.

$\mathbb{S}ol^3\times\mathbb{E}^1$-manifolds

We show that $\mathbb{S}ol^3\times\mathbb{E}^1$-manifolds are Seifert fibred, with general fibre the torus, and base one of the seven flat 2-orbifolds $T, Kb, \mathbb{A}, \mathbb{M}b, S(2,2,2,2), P(2,2)$ or $\mathbb{D}(2,2)$, and outline a classification of such 4-manifolds.

小Seifert纤维空间的填充性

Fillability of small Seifert fibered spaces

本文研究小Seifert纤维空间上切触结构的填充性，通过分析平面开书的单值因子化，刻画了$M(-1;r_1,r_2,r_3)$上零扭转切触结构中的可填充结构。

非纤维化强拟正链环与紧性

Non-fibered strongly quasipositive links and tightness

本文利用部分开书研究非纤维化链环中强拟正性与紧接触结构的关系，证明强拟正蕴含紧性，并构造反例说明逆命题不成立。

Classical invariants of Legendrian knots in the 3-dimensional torus

All knots in $R^3$ possess Seifert surfaces, and so the classical Thurston-Bennequin and rotation (or Maslov) invariants for Legendrian knots in a contact structure on $R^3$ can be defined. The definitions extend easily to null-homologous knots in any $3$-manifold $M$ endowed with a contact structure $ξ$. We generalize the definition of Seifert surfaces and use them to define these invariants for all Legendrian knots, including those that are not null-homologous, in a contact structure on the $3$-torus $T^3$. We show how to compute the Thurston-Bennequin and rotation invariants in a tight orie…

具有两个可填充接触结构的Brieskorn球面

Brieskorn spheres with two fillable contact structures

本文基于对Seifert纤维空间上负扭转紧致接触结构的分类，给出了所有最多携带两个辛可填充结构的Brieskorn球面的完整列表，这些结构在同伦意义下与给定的定向相容。

Explicit horizontal open books on some plumbings

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, we describe horizontal open books on some Seifert fibered 3--manifolds. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3--manifolds supported by planar open books. As a cons…

Tight Contact Structures via Admissible Transverse Surgery

We investigate the line between tight and overtwisted for surgeries on fibred transverse knots in contact 3-manifolds. When the contact structure $ξ_K$ is supported by the fibred knot $K \subset M$, we obtain a characterisation of when negative surgeries result in a contact structure with non-vanishing Heegaard Floer contact class. To do this, we leverage information about the contact structure $ξ_{\overline{K}}$ supported by the mirror knot $\overline{K} \subset -M$. We derive several corollaries about the existence of tight contact structures, L-space knots outside $S^3$, non-planar contact …

Planar open books, monodromy factorizations, and symplectic fillings

We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl's theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on L(p,1) has a unique filling, and describe fillable and non-fillable tight contact structures on certain Seifert fibered spaces.

Heegaard Floer接触不变量的帽子版和加号版不等价

The hat and plus version of the Heegaard Floer contact invariant are not equivalent

研究揭示了3流形上接触结构的特性，展示存在紧致且零扭转的接触结构，其中Heegaard Floer不变量非零而加号不变量为零。

Open book decompositions of fibre sums in contact topology

In the present paper we describe compatible open books for the fibre connected sum along binding components of open books, as well as for the fibre connected sum along multi-sections of open books. As an application the first description provides simple ways of constructing open books supporting all tight contact structures on $T^3$, recovering a result by van Horn-Morris, as well as an open book supporting the result of a Lutz twist along a binding component of an open book, recovering a result by Ozbagci--Pamuk.

Compatible contact structures of fibered positively-twisted graph multilinks in the 3-sphere

We study compatible contact structures of fibered, positively-twisted graph multilinks in the 3-sphere and prove that the contact structure of such a multilink is tight if and only if the orientations of its link components are all consistent with or all opposite to the orientation of the fibers of the Seifert fibrations of that graph multilink. As a corollary, we show that the compatible contact structures of the Milnor fibrations of real analytic germs of the form (f\bar g,O) are always overtwisted.

Legendrian ribbons in overtwisted contact structures

We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies $\sel(K,S)=-χ(S)$. In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class.

Ozsvath-Szabo invariants and tight contact three-manifolds, I

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact structures for every r not= 2g_s(K)-1, where g_s(K) is the slice genus of K. This implies, in particular, that the Brieskorn spheres -Sigma(2,3,4) and -Sigma(2,3,3) carry tight, positive contact structures. As an application of our main result we show that for each m in N there exists a Seifert fibered rational homology 3-sphere M_m carrying at lea…

Exceptional Legendrian torus knots

We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact structures on certain Seifert fibred manifolds with boundary allow us to place upper bounds on the number of tight contact structures on the complements of torus knots; the classification of exceptional realisations of these torus knots is then achieved by exhibiting sufficiently many realisations in terms of contact surgery diagrams. We also discuss a couple of ge…

A cut-and-paste approach to contact topology

Contact structures on 3-manifolds are analyzed by decomposing the manifold along convex surfaces. Background results of Giroux, Eliashberg, Colin, and Honda are discussed with an emphasis on examples. Convex decompositions are then used to give a new proof of the Gabai-Eliashberg-Thurston Theorem on the existence of universally tight contact structures and also to study the contact topology of a space in the presence or absence of tori. Classification of tight contact structures on fibred manifolds and related open questions are also discussed. This paper is based on a series of talks given at…

紧接触3-球面的环面结过滤嵌入接触同调

Torus knot filtered embedded contact homology of the tight contact 3-sphere

本文推广了结过滤嵌入接触同调以处理具有有理旋转数的退化结，并利用 Morse-Bott 方法和 Seiberg-Witten 理论计算了紧接触 3-球面上 T(2,q) 环面结的过滤嵌入接触同调，为后续正 T(p,q) 结计算及 Reeb 动力学应用奠定基础。

大圆纤维化与奇数维球面上的切触结构

Great Circle Fibrations and Contact Structures on Odd-Dimensional Spheres

本文证明从5-球面开始，所有奇数维球面都存在与纤维正交的超平面分布非切触结构的光滑大圆纤维化，打破了3-球面中的已知结论。

Structures de contact sur les varietes fibrees en cercles au-dessus d'une surface

In this paper, we study the global behaviour of contact structures on oriented manifolds V which are circle bundles over a closed orientable surface S of genus g>0. We establish in particular contact analogs of a number of classical results about foliations due to Milnor, Wood, Thurston, Matsumoto, and Ghys. In Section~1, we prove that V carries a (positive) contact structure transverse to the fibers if and only if the Euler number of the fibration is less or equal to 2g-2. In Section~2, we show that, for any contact structure $ξ$ on V, one of the following properties holds: either $ξ$ is isot…

Great circle fibrations and contact structures on the 3-sphere

Given any smooth fibration of the unit 3-sphere by great circles, we show that the distribution of 2-planes orthogonal to the great circle fibres is a tight contact structure, a fact well known in the special case of the Hopf fibrations. The proof expresses hypothesis and conclusion as differential inequalities involving functions on disks transverse to the fibres, and shows that one inequality implies the other.

Some applications of Menke's JSJ decomposition for symplectic fillings

We apply Menke's JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over $S^2$, we reduce the problem of classifying exact symplectic structures to the …

Contact structures, deformations and taut foliations

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields is in general not path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seif…

Surface singularities and planar contact structures

We prove that if a contact 3-manifold admits an open book decomposition of genus 0, a certain intersection pattern cannot appear in the homology of any of its minimal symplectic fillings, and moreover, fillings cannot contain symplectic surfaces of positive genus. Applying these obstructions to canonical contact structures on links of normal surface singularities, we show that links of isolated singularities of surfaces in the complex 3-space are planar only in the case of $A_n$-singularities. In general, we characterize completely planar links of normal surface singularities (in terms of thei…

Strong fillability and the Weinstein conjecture

Extending work of Chen, we prove the Weinstein conjecture in dimension three for strongly fillable contact structures with either non-vanishing first Chern class or with strong and exact filling having non-trivial canonical bundle. This implies the Weinstein conjecture for certain Stein fillable contact structures obtained by the Eliashberg-Gompf construction.For example we prove the Weinstein conjecture for the Brieskorn homology spheres $Σ(2,3,6n-1)$, $n\geq2$, oriented as the boundary of the corresponding Milnor fibre. Furthermore, for tight contact structures on odd lens spaces, non-contra…