On the topology of the space of contact structures on torus bundles

Knot contact homology detects cabled, composite, and torus knots

Knot contact homology is an invariant of knots derived from Legendrian contact homology which has numerous connections to the knot group. We use basic properties of knot groups to prove that knot contact homology detects every torus knot. Further, if the knot contact homology of a knot is isomorphic to that of a cable (respectively composite) knot, then the knot is a cable (respectively composite).

Tight, not semi--fillable contact circle bundles

Extending our earlier results, we prove that certain tight contact structures on circle bundles over surfaces are not symplectically semi--fillable, thus confirming a conjecture of Ko Honda.

Canonical contact unit cotangent bundle

We describe an explicit open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of a compact surface.

3 - 环面的接触手术数

Contact Surgery Numbers of the 3-torus

研究3 - 环面接触结构的接触手术数，证明沿Legendrian博罗梅安环手术得过度扭曲结构且有无限多个非接触同胚结构，给出三分量链环相关障碍，还用特定方法证明\((\mathbb{T}^3, \xi_1)\)斯坦填充的总辛同调不为零。

On the capacity of Lagrangians in the cotangent disc bundle of the torus

The paper is wihdrawn due to a critical error in the argument using the spectral sequence

Exact Lagrangians in the cotangent bundle of a sphere and a torus

It is known that any closed, exact Lagrangian in the cotangent bundle of a closed, smooth manifold is of the same homotopy type as the zero section. In this paper, we give a Fukaya-theoretic proof of this fact for the sphere and torus to review and demonstrate some of the homological algebra techniques in symplectic geometry.

Existence of symplectic structures on torus bundles over surfaces

Let E be the total space of a locally trivial torus bundle over the surface Σ_g of genus g>1. Using the Seiberg--Witten theory and spectral sequences we prove that E carries a symplectic structure if and only if the homology class of the fiber [T^2] is nonzero in H_2(E,R).

On the contact mapping class group of Legendrian circle bundles

In this paper, we determine the group of contact transformations modulo contact isotopies for Legendrian circle bundles over closed surfaces of nonpositive Euler characteristic. These results extend and correct those presented by the first author in a former work. The main ingredient we use is connectedness of certain spaces of embeddings of surfaces into contact 3-manifolds. In the third section, this connectedness question is studied in more details with a number of (hopefully instructive) examples.

Legendrian torus knots in lens spaces

In this note, we first classify all topological torus knots lying on the Heegaard torus in lens spaces, and then we study Legendrian representatives of these knots. We classify oriented positive Legendrian torus knots in the universally tight contact structures on the lens spaces up to contactomorphism.

The fundamental group of the space of contact structures on the 3-torus

We show that the fundamental group of the space of contact structures on the 3-torus (based at the standard contact structure) is isomorphic to the integers.

环面上纽结的紧手术

Tight surgeries on torus knots

本文对沿任意环面上纽结手术所得流形上的紧接触结构完成分类，还分类了相关曲面丛的接触结构，研究其辛可填充性并反驳了一项相关猜想。

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。

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…

Symplectic dynamics of contact isotropic torus complements

We determine the homotopy type of isotropic torus complements in closed contact manifolds in terms of Reeb dynamics of special contact forms. For that we utilize holomorphic curve techniques known from symplectic field theory as Gromov-Hofer compactness and localized transversality on non-compact contact manifolds.

On Stein fillings of contact torus bundles

We consider a large family F of torus bundles over the circle, and we use recent work of Li--Mak to construct, on each Y in F, a Stein fillable contact structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first Chern class and first Betti number, (ii) if Y in F is elliptic then all Stein fillings of (Y,C) are pairwise diffeomorphic and (iii) if Y in F is parabolic or hyperbolic then all Stein fillings of (Y,C) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y in F we exhibit non-homotopy…

非凸马瑟理论与托氏圆柱的康利猜想

Non-convex Mather's theory and the Conley conjecture on the cotangent bundle of the torus

本文通过辛同调化方法证明了非凸马瑟理论中关于圆柱的康利猜想及周期轨道的存在性。

Non-fillable invariant contact structures on principal circle bundles and left-handed twists

We define symplectic fractional twists, which generalize Dehn twists, and use these in open books to investigate contact structures. The resulting contact structures are invariant under a circle action, and share several similarities with the invariant contact structures that were studied by Lutz and Giroux. We show that left-handed fractional twists often give rise to non-fillable contact manifolds. These manifolds are in fact "algebraically overtwisted", yet they do not seem to contain bLobs, nor are they directly related to negative stabilizations. We also show that the Weinstein conjecture…

Berge duals and universally tight contact structures

Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in $S^3$ that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology class as the dual of a torus knot.

Stein and Weinstein structures on disk cotangent bundles of surfaces

In 1998, Gompf described a Stein domain structure on the disk cotangent bundle of any closed surface S, by a Legendrian handlebody diagram. We prove that Gompf's Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorphic to the unit cotangent bundle of S equipped with its canonical contact structure. As a corollary, we obtain a surgery diagram for the canonical contact structure on the unit cotangent bundle of S.

Uniqueness of the contact structure approximating a foliation

According to a theorem of Eliashberg and Thurston a $C^2$-foliation on a closed 3-manifold can be $C^0$-approximated by contact structures unless all leaves of the foliation are spheres. Examples on the 3-torus show that every neighbourhood of a foliation can contain non-diffeomorphic contact structures. In this paper we show uniqueness up to isotopy of the contact structure in a small neighbourhood of the foliation when the foliation has no torus leaf and is not a foliation without holonomy on parabolic torus bundles over the circle. This allows us to associate invariants from contact topolog…

A convexity theorem for torus actions on contact manifolds

We show that the cone associated with a moment map for an action of a torus on a contact compact connected manifold is a convex polyhedral cone and that the moment map has connected fibers provided the dimension of the torus is bigger than 2 and that no orbit is tangent to the contact distribution. This may be considered as a version of the Atiyah - Guillemin - Sternberg convexity theorem for torus actions on symplectic cones and as a direct generalization of the convexity theorem of Banyaga and Molino for completely integrable torus actions on contact manifolds.

Symplectic torus bundles and group extensions

Symplectic torus bundles $ξ:T^{2}\to E\to B$ are classified by the second cohomology group of $B$ with local coefficients $H_{1}(T^{2})$. For $B$ a compact, orientable surface, the main theorem of this paper gives a necessary and sufficient condition on the cohomology class corresponding to $ξ$ for $E$ to admit a symplectic structure compatible with the symplectic bundle structure of $ξ$ : namely, that it be a torsion class. The proof is based on a group-extension-theoretic construction of J. Huebschmann (Sur les premieres differentielles de la suite spectrale cohomologique d'une extension de …

Contact fiber bundles

We define contact fiber bundles and investigate conditions for the existence of contact structures on the total space of such a bundle. The results are analogous to minimal coupling in symplectic geometry. The two applications are construction of K-contact manifolds generalizing Yamazaki's fiber join construction and a cross-section theorem for contact moment maps

Conormal bundles, contact homology and knot invariants

We summarize recent work on a combinatorial knot invariant called knot contact homology. We also discuss the origins of this invariant in symplectic topology, via holomorphic curves and a conormal bundle naturally associated to the knot.

Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle

We show that in many examples the non-displaceability of Lagrangian submanifolds by Hamiltonian isotopy can be proved via Lagrangian Floer cohomology with non-unitary line bundle. The examples include all monotone Lagrangian torus fibers in toric Fano manifold (which was also proven by Entov and Polterovich via the theory of symplectic quasi-states), some non-monotone Lagrangian torus fibers. We also extend the results by Oh and the author about the computations of Floer cohomology of Lagrangian torus fibers to the case of all toric Fano manifolds, removing the convexity assumption in the prev…

Cosphere bundle reduction in contact geometry

We extend the theorems concerning the equivariant symplectic reduction of the cotangent bundle to contact geometry. The role of the cotangent bundle is taken by the cosphere bundle. We use Albert's method for reduction at zero and Willett's method for non-zero reduction. We provide examples for both cases.

关于环面作用的锥形变换的渐近等距嵌入的评注

Remarks on asymptotic isometric embeddings of conic transforms for torus actions

本文研究 Hamilton 全纯环面作用下量子化线丛对偶圆丛的几何，利用等变 Szegő 投影算子构造与不可约权射线相关的锥形变换的渐近嵌入。

Contact structures on principal circle bundles

We describe a necessary and sufficient condition for a principal circle bundle over an even-dimensional manifold to carry an invariant contact structure. As a corollary it is shown that all circle bundles over a given base manifold carry an invariant contact structure, only provided the trivial bundle does. In particular, all circle bundles over 4-manifolds admit invariant contact structures. We also discuss the Bourgeois construction of contact structures on odd-dimensional tori in this context, and we relate our results to recent work of Massot, Niederkrueger and Wendl on weak symplectic fil…

Regular Jacobi Structures and Generalized Contact Bundles

A Jacobi structure $J$ on a line bundle $L\to M$ is weakly regular if the sharp map $J^\sharp : J^1 L \to DL$ has constant rank. A generalized contact bundle with regular Jacobi structure possess a transverse complex structure. Paralleling the work of Bailey in generalized complex geometry, we find condition on a pair consisting of a regular Jacobi structure and an transverse complex structure to come from a generalized contact structure. In this way we are able to construct interesting examples of generalized contact bundles. As applications: 1) we prove that every 5-dimensional nilmanifold i…

环面丛 Liouville 域是稳定 Weinstein 的

Torus bundle Liouville domains are stably Weinstein

本文提出了简化 Liouville 向量场动力学的局部操作，并以此证明了 Huang 构造的某些 Liouville-but-not-Weinstein 域是稳定 Weinstein 的。

Contact homology of $S^1$-bundles over some symplectically reduced orbifolds

In this paper, we compute contact homology of some quasi-regular contact structures, which admit Hamiltonian actions of Reeb type of Lie groups. We will discuss the toric contact case, (where the torus is of Reeb type), and the case of homogeneous contact manifolds. In both of these cases the quotients by the Reeb action are Kähler and admit perfect Morse-Bott functions via the moment map. It turns out that the contact homology depends only on the homology of the symplectic base and the bundle data of the contact manifolds as a circle bundle over the base. Moreover we can identify the relevant…

接触线丛、叶状结构与可积性

Contact line bundles, foliations, and integrability

本研究利用接触线丛截面空间的Jacobi结构构建接触系统非交换可积性，可统一处理两类接触流形的接触系统，还可用于研究耗散哈密顿系统。

A New Characterization of the Clifford Torus

In this paper we introduce the notion of contact angle for an immersed surface in three dimensional sphere. We deduce formulas for the Laplacian and for the Gaussian curvature, and we classify minimal surfaces in $S^3$ with constant contact angle. Also, we give an example of a minimal surface in $S^3$ with non constant contact angle.

奇异余切球丛约化

Singular cosphere bundle reduction

本文研究动量映射零值下余切球丛的奇异切触约化，提出一种更适配丛结构的更精细奇异商分层，其层由余切球丛及对应切触分量的余迷向、勒让德子流形组成。

On the Equivalence Problem for Toric Contact Structures on S^3-bundles over S^2$

We study the contact equivalence problem for toric contact structures on $S^3$-bundles over $S^2$. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To find inequivalent toric contact structures that are contact equivalent, we show that the corresponding 3-tori belong to distinct conjugacy classes in the contactomorphism group. To show that two toric contact structures with the same first Chern class are contact inequivalent, we …

The Local Structure of Generalized Contact Bundles

Generalized contact bundles are odd dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.

Legendrian切触瞬子上同调及其在一Jet丛上的谱不变量

Legendrian contact instanton cohomology and its spectral invariants on the one-jet bundle

本文发展了Legendrian链相关哈密顿扰动切触作用泛函的Floer型椭圆Morse理论，构造了Legendrian切触瞬子上同调及其谱不变量，涵盖了此前余切丛上的相关Lagrangian相交理论成果。

环面与球面上的接触结构和贝尔特拉米场

Contact structures and Beltrami fields on the torus and the sphere

本文在3-球面和3-环面上构造了新的显式紧和过度扭转接触结构，证明了弱相容度量下接触球面定理不成立，并建立了紧接触结构的几何刚性。

Topology of holomorphic Lefschetz pencils on the four-torus

In this paper we discuss topological properties of holomorphic Lefschetz pencils on the four-torus. Relying on the theory of moduli spaces of polarized abelian surfaces, we first prove that, under some mild assumption, the (smooth) isomorphism class of a holomorphic Lefschetz pencil on the four-torus is uniquely determined by its genus and divisibility. We then explicitly give a system of vanishing cycles of the genus-3 holomorphic Lefschetz pencil on the four-torus due to Smith, and obtain those of holomorphic pencils with higher genera by taking finite unbranched coverings. One can also obta…

Generalized Contact Bundles

In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators, it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.

Klein 瓶单位余切丛的嵌入切触同调

Embedded contact homology of the unit cotangent bundle of the Klein bottle

本文给出了 Klein 瓶单位余切丛嵌入切触同调链复形的组合描述，并利用 pseudoholomorphic curves 建立了环面区域辛嵌入的阻碍定理，进而计算出 $D^*K$ 的 Gromov 宽度。

非退化空隙接触形式在预量化丛上的周期轨道

Periodic orbits of non-degenerate lacunary contact forms on prequantization bundles

研究非退化空隙接触形式在预量化丛上的周期轨道数量及其与辛流形的拓扑性质的关系。

A novel approach to contact Hamiltonians and contact Hamilton-Jacobi theory

We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact manifold M itself but sections of a line bundle over M or, equivalently, 1-homogeneous functions on a certain GL(1,R)-principal bundle P\to M which is equipped with a homogeneous symplectic form ω. In other words, our understanding of contact geometry is that it is not an `odd-dimensional cousin' of symplectic geometry but rather a part of the latter, namely `…

Asymptotics of Szegö kernels under Hamiltonian torus actions

Let $X$ be the circle bundle associated to a positive line bundle on a complex projective (or, more generally, compact symplectic) manifold. The Tian-Zelditch expansion on $X$ may be seen as a local manifestation of the decomposition of the (generalized) Hardy space $H(X)$ into isotypes for the $S^1$-action. More generally, given a compatible action of a compact Lie group, and under general assumptions guaranteeing finite dimensionality of isotypes, we may look for asymptotic expansions locally reflecting the equivariant decomposition of $H(X)$ over the irreducible representations of the group…

接触几何力学：Tulczyjew三元组

Contact geometric mechanics: the Tulczyjew triples

该研究将经典Tulczyjew三元组推广至接触流形，以线丛一阶射丛的典范接触结构替代余切丛辛结构，定义接触哈密顿与拉格朗日量的动力学，还研究了接触勒让德映射与变换并给出实例。

Remarks on Contact and Jacobi Geometry

We present an approach to Jacobi and contact geometry that makes many facts, presented in the literature in an overcomplicated way, much more natural and clear. The key concepts are Kirillov manifolds and linear Kirillov structures, i.e., homogeneous Poisson manifolds and, respectively, homogeneous linear Poisson manifolds. The difference with the existing literature is that the homogeneity of the Poisson structure is related to a principal ${\rm GL}(1,{\mathbb R})$-bundle structure on the manifold and not just to a vector field. This allows for working with Jacobi bundle structures on nontriv…