Homotopy groups of K-contact toric manifold

A Note on Toric Contact Geometry

After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective action of a torus whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we use this fact together with a recent symplectic orbifold version of Delzant's theorem due to Lerman and Tolman [LT] to show that every such compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.

The momentum polytope of actions on contact manifolds via toric varieties

Due to a previous result which states that contact varieties are isomorphic to certain varieties, the momentum polytopes of contact manifolds are convex.

Combinatorial embedded contact homology for toric contact manifolds

Computing embedded contact homology (ECH) and related invariants of certain toric 3-manifolds (in the sense of Lerman) has led to interesting new results in the study of symplectic embeddings. Here, we give a combinatorial formulation of ECH chain complexes for general toric contact 3-manifolds. As a corollary, we prove Conjecture A.3 from [Hutchings14].

On displaceability of pre-Lagrangian fibers in contact toric manifolds

In this note we analyze displaceability of pre-Lagrangian toric fibers in contact toric manifolds. While every symplectic toric manifold contains at least one non-displaceable Lagrangian toric fiber and infinitely many displaceable ones, we show that this is not the case for contact toric manifolds. More precisely, we prove that for the contact toric manifolds $\mathbb{S}^{2d-1} (d\geq 2)$ and $\mathbb{T}^k \times \mathbb{S}^{2d+k-1} (d \geq 1)$ all pre-Lagrangian toric fibers are displaceable, and that for all contact toric manifolds for which the toric action is free, except possibly non-tri…

Spherical contact toric manifolds

Let $(M, α)$ be a $2n+1$-dimensional connected compact contact toric manifold of Reeb type. Suppose the contact form $α$ is regular, we find conditions under which $M$ is homeomorphic to $S^{2n+1}$.

环面接触流形上的等变指标

Equivariant Index on Toric Contact Manifolds

本文计算了Reeb型紧致环面接触流形上扭曲水平Dolbeault算子的等变指标，利用好锥条件证明了符号局部化，并得到了Atiyah-Bott-Lefschetz型公式及基于矩锥格点求和的指标表达式。

Symplectic fillability of toric contact manifolds

According to Lerman, compact connected toric contact 3-manifolds with a non-free toric action whose moment cone spans an angle greater than $π$ are overtwisted, thus non-fillable. In contrast, we show that all compact connected toric contact manifolds in dimension greater than three are weakly symplectically fillable and most of them are strongly symplectically fillable. The proof is based on the Lerman's classification of toric contact manifolds and on our observation that the only contact manifolds in higher dimensions that admit free toric action are the cosphere bundle of $T^d, d\geq3$ $(T…

Completely Integrable Contact Hamiltonian Systems and Toric Contact Structures on $S^2\times S^3$

I begin by giving a general discussion of completely integrable Hamiltonian systems in the setting of contact geometry. We then pass to the particular case of toric contact structures on the manifold $S^2\times S^3$. In particular we give a complete solution to the contact equivalence problem for a class of toric contact structures, $Y^{p,q}$, discovered by physicists by showing that $Y^{p,q}$ and $Y^{p',q'}$ are inequivalent as contact structures if and only if $p\neq p'$.

Geodesic flows and contact toric manifolds

These are notes for a course on contact manifolds and torus actions delivered at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems at Centre de Recerca Matemàtica in Barcelona in July 2001. To be published by Birkhauser.

容许非自由环面作用的切触3-流形

Contact 3-manifolds that admit a non-free toric action

本研究基于Lerman的工作，通过显式描述完成切触环面3-流形的切触同胚分类，并进一步分类了可作为球面上线形管道凹边界的3-流形的所有切触结构，相关结果受MNRSTW25工作启发。

Translated points for contactomorphisms of prequantization spaces over monotone symplectic toric manifolds

We prove a version of Sandon's conjecture on the number of translated points of contactomorphisms for the case of prequantization bundles over certain closed monotone symplectic toric manifolds. Namely we show that any contactomorphism of such a prequantization bundle lying in the identity component of the contactomorphism group possesses at least $N$ translated points, where $N$ is the minimal Chern number of the symplectic toric manifold. The proof relies on the theory of generating functions coupled with equivariant cohomology, whereby we adapt Givental's approach to the Arnold conjecture f…

允许环面可积测地流的3维流形

3-Manifolds admitting toric integrable geodesic flows

本文利用余球面丛在环面可积时具有的接触环面流形结构，通过比较Betti数，得到紧连通3维流形上测地流环面可积的必要条件。

Q-Gorenstein环面接触流形的接触不变量、Ehrhart多项式与Chen-Ruan上同调

Contact invariants of Q-Gorenstein toric contact manifolds, the Ehrhart polynomial and Chen-Ruan cohomology

本文研究Q-Gorenstein环面接触流形的柱形接触同调不变量，建立其与环面图的Ehrhart（拟）多项式、对应环面孤立奇点的crepant环面轨形消解的Chen-Ruan上同调，以及生成该流形的单调环面辛轨形底空间的Chen-Ruan上同调的关联。

加权射影空间的同调镜像对称与Morse同伦

Homological mirror symmetry for weighted projective spaces and Morse Homotopy

本文将环面流形上的加权Morse同伦范畴与凝聚层导出范畴的同调镜像对称研究推广至环面轨形，讨论了加权射影空间的对应版本同调镜像对称。

接触柔韧性与刚性 for toric Gorenstein 预量化及 toric 图的 Ehrhart 理论

Contact flexibility and rigidity for toric Gorenstein prequantizations and Ehrhart theory of toric diagrams

本文研究 toric Gorenstein 接触流形的柔韧性与刚性，探讨其与 Ehrhart 多项式及接触 betti 数的关系，证明了某些预量化流形的唯一性和刚性。

边界为透镜空间的环面域的辛嵌入

Symplectic embeddings of toric domains with boundary a lens space

本研究给出凹透镜空间的嵌入切触复形组合描述，定义带奇点的广义凹环面域，利用ECC计算其ECH容量，并给出可恢复该容量的辛流形填充方式。

切触环面流形的基本群

The fundamental groups of contact toric manifolds

本文研究连通紧致切触环面流形的基本群，证明Reeb型这类流形的基本群为有限循环群，并给出从矩映射像求其阶的方法。

On the mean Euler characteristic of Gorenstein toric contact manifolds

We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, i.e. a good toric contact manifold with zero first Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and Dais, is the following: twice the mean Euler characteristic of a Gorenstein toric contact manifold is equal to the Euler characteristic of any crepant toric symplectic filling, i.e. any toric symplectic filling with zero first Chern class.

Cohomology rings of good contact toric manifolds

A good contact toric manifold $M$ is determined by its moment cone $C$. We compute the equivariant cohomology ring with $\Z$ coefficient of $M$ in terms of the combinatorial data of $C$. Then under a smoothness criterion on the cone $C$, we compute the singular cohomology ring with $\Z$ coefficient of $M$ in terms of the combinatorial data of $C$.

Symplectic toric stratified spaces with isolated singularities

The goal of this paper is to classify symplectic toric stratified spaces with isolated singularities. This extends a result of Burns, Guillemin, and Lerman which carries out this classification in the compact connected case. In making this classification, it is necessary to classify symplectic toric cones. Via a well-known equivalence between symplectic toric cones and contact toric manifolds, this allows for the classification of contact toric manifolds as well, extending Lerman's classification of compact connected contact toric manifolds.

Contact homology and homotopy groups of the space of contact structures

Using contact homology, we reobtain some recent results of Geiges and Gonzalo about the fundamental group of the space of contact structures on some 3-manifolds. We show that our techniques can be used to study higher dimensional contact manifolds and higher order homotopy groups.

Contact homology of good toric contact manifolds

In this paper we show that any good toric contact manifold has well defined cylindrical contact homology and describe how it can be combinatorially computed from the associated moment cone. As an application we compute the cylindrical contact homology of a particularly nice family of examples that appear in the work of Gauntlett-Martelli-Sparks-Waldram on Sasaki-Einstein metrics. We show in particular that these give rise to a new infinite family of non-equivalent contact structures on $S^2 \times S^{3}$ in the unique homotopy class of almost contact structures with vanishing first Chern class…

Homological mirror symmetry of $\mathbb{F}_1$ via Morse homotopy

This is a sequel to our paper arXiv:2008.13462, where we proposed a definition of the Morse homotopy of the moment polytope of toric manifolds. Using this as the substitute of the Fukaya category of the toric manifolds, we proved a version of homological mirror symmetry for the projective spaces and their products via Strominger-Yau-Zaslow construction of the mirror dual Landau-Ginzburg model. In this paper we go this way further and extend our previous result to the case of the Hirzebruch surface $\mathbb{F}_1$.

On contact invariants of non-simply connected Gorenstein toric contact manifolds

The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper we show that the same explicit method can be used to compute Conley-Zehnder indices of non-contractible periodic Reeb orbits. Under appropriate conditions, the (finite) number of such orbits in a given free homotopy class and with a given index is a contact invariant of the underl…

有理直纹面上 Hamilton 圆周作用的中心化子

Centralizers of Hamiltonian circle actions on rational ruled surfaces

本文研究两类有理直纹面上 Hamilton $S^1$ 作用的等变辛微分同胚群，通过近复结构分层及 Delzant、Karshon 分类确定其同伦型为环面或两环面的同伦推出。

接触复曲面结构的性质与线性拼接的凹边界

Properties of contact toric structures and concave boundaries of linear plumbings

研究具有凹接触边界的球面上辛圆盘束的线性拼接，用复曲面几何工具确定接触结构紧或过度扭曲情况，开发工具包计算线性拼接凹边界的ECH代数挠率和接触不变量。

凹型辛流形的toric填充

Concave symplectic toric fillings

本文研究了接触toric3维流形的凹型辛填充结构，证明了其存在无限多个互不equivariant辛同构且不通过吹胀操作相关的填充。

Homological mirror symmetry of $\mathbb{C}P^n$ and their products via Morse homotopy

We propose a way of understanding homological mirror symmetry when a complex manifold is a smooth compact toric manifold. So far, in many example, the derived category $D^b(coh(X))$ of coherent sheaves on a toric manifold $X$ is compared with the Fukaya-Seidel category of the Milnor fiber of the corresponding Landau-Ginzburg potential. We instead consider the dual torus fibration $π:M \to B$ of the complement of the toric divisors in $X$, where $\bar{B}$ is the dual polytope of the toric manifold $X$. A natural formulation of homological mirror symmetry in this set-up is to define $Fuk(\bar{M}…

Almost contact 5-folds are contact

We prove that every homotopy class of almost contact structures on a closed 5-dimensional manifold admits a contact structure.

Vortex invariants and toric manifolds

We consider the symplectic vortex equations for a linear Hamiltonian torus action. We show that the associated genus zero moduli space itself is homotopic (in the sense of a homotopy of regular G-moduli problems) to a toric manifold with combinatorial data directly obtained from the original torus action. This allows to view the wall crossing formula of Cieliebak and Salamon for the computation of vortex invariants as a consequence of a generalized Jeffrey-Kirwan localization formula for integrals over symplectic quotients.

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…

Topology of spaces of equivariant symplectic embeddings

We compute the homotopy type of the space of T^n-equivariant symplectic embeddings from the standard 2n-dimensional ball of some fixed radius into a 2n-dimensional symplectic-toric manifold M, and use this computation to define a Z-valued step function on the positive real line which is an invariant of the symplectic-toric type of M. We conclude with a discussion of the partially equivariant case of this result.

一种切触同伦型

A contact homotopy type

本文将global Kuranishi charts调整至切触设定，基于Reeb orbits和伪全纯建筑模空间为非退化contact manifold构建流范畴，该构造提升了contact homology并具有自然性。

Non-zero contact and Sasakian reduction

We complete the reduction of Sasakian manifolds with the non-zero case by showing that Willett's contact reduced space is compatible with the Sasakian structure. We then prove the compatibility of the non-zero Sasakian (in particular, contact) reduction with the reduction of the Kähler (in particular, symplectic) cone. We provide examples obtained by toric actions on Sasakian spheres and make some comments concerning the curvature of the quotients.

凹奇异环面域的 ECH 容量

ECH capacities of concave singular toric domains

本文推广四维环面域至边界为透镜空间 L(n,1) 的奇异环面域，并计算一类凹奇异环面域的 ECH 容量，结果可用于研究其去奇异化空间中的嵌入问题。

Kahler-Sasaki geometry of toric symplectic cones in action-angle coordinates

In the same way that a contact manifold determines and is determined by a symplectic cone, a Sasaki manifold determines and is determined by a suitable Kahler cone. Kahler-Sasaki geometry is the geometry of these cones. This paper presents a symplectic action-angle coordinates approach to toric Kahler geometry and how it was recently generalized, by Burns-Guillemin-Lerman and Martelli-Sparks-Yau, to toric Kahler-Sasaki geometry. It also describes, as an application, how this approach can be used to relate a recent new family of Sasaki-Einstein metrics constructed by Gauntlett-Martelli-Sparks-W…

Equivariant symplectic homology, linearized contact homology and the Lagrangian capacity

We establish computational results concerning the Lagrangian capacity from "Cieliebak and Mohnke - Punctured holomorphic curves and Lagrangian embeddings". More precisely, we show that the Lagrangian capacity of a 4-dimensional convex toric domain is equal to its diagonal. The proof involves comparisons between the Lagrangian capacity, the McDuff-Siegel capacities from "McDuff and Siegel - Symplectic capacities, unperturbed curves, and convex toric domains", and the Gutt-Hutchings capacities from "Gutt and Hutchings - Symplectic capacities from positive S1-equivariant symplectic homology". Wor…

Constructions in Sasakian Geometry

We describe various constructions in Sasakian geometry. First we generalize the join construction of the first two authors to arbitrary Sasakian manifolds. We then give several examples, including ones which prove the existence of Sasakian-Einstein metrics on manifolds homeomorphic to $S^2\times S^5.$ Then we use a generalization of the join construction due to Lerman, namely contact fibre bundles, to give a theorem constructing toric Sasakian structures. Finally, we explicitly construct regular toric Sasakian structures on all simply connected regular contact manifolds in dimension five.

Five dimensional K-contact manifolds of rank 2

A K-contact manifold is a smooth manifold M with a contact form whose Reeb flow preserves a Riemannian metric on M. Main examples are Sasakian manifolds. Our results in this paper are four results i), ii), iii) and iv) below obtained by the application of the Morse theory to the contact moment maps on closed 5-dimensional K-contact manifolds of rank 2. 5-dimensional K-contact manifolds of rank 2 have the lowest dimension among K-contact manifolds with nontrivial contact moment maps which are not toric. i) Correspondence between the isomorphism classes of K-contact manifolds and graphs of isotr…

The topology of a subspace of the Legendrian curves in a closed contact 3-manifold

In this paper we study a subspace of the space of Legendrian loops and we show that the injection of this space into the full loop space is an S1-equivariant homotopy equivalence. This space can be also seen as the space of zero Maslov index Legendrian loops and it shows up as a suitable space of variations in contact form geometry.

An absolute grading on Heegaard Floer homology by homotopy classes of oriented 2-plane fields

For a closed oriented 3-manifold Y, we define an absolute grading on the Heegaard Floer homology groups of Y by homotopy classes of oriented 2-plane fields. We show that this absolute grading refines the relative one and that it is compatible with the maps induced by cobordisms. We also prove that if ξ is a contact structure on Y, then the grading of the contact invariant c(ξ) is the homotopy class of ξ.

Higher Maslov Indices

We define Maslov--type indices associated to contact and symplectic transformation groups. There are two such families of indices. The first class of indices are maps from the homotopy groups of the contactomorphism or symplectomorphism group to a quotient of the integers. These are based on a generalization of the Maslov index. The second class of indices are maps from the homotopy groups of the space of contact structures or the space of cohomologous symplectic forms to the homotopy groups of a simple homogeneous space. We provide a detailed construction and describe some properties of these…

关于非序流形中切触微分同胚环与勒让德子流形环的拓扑

On the topology of loops of contactomorphisms and Legendrians in non-orderable manifolds

本文研究了非序闭切触流形的切触微分同胚环与勒让德子流形环空间的全局拓扑，通过正性度量进行过滤，并证明了其同伦群为正环子空间同伦群的子群。

The homotopy Lie algebra of symplectomorphism groups of 3-fold blow-ups of the projective plane

By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples, the rational cohomology rings of symplectomorphism groups are finitely generated. In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional. Moreover, we explain how the topology is generated by…

三维中的紧接触结构空间

Spaces of tight contact structures in dimension 3

该研究引入微纤维化方法，证明紧接触结构空间的弱同伦型在接触0-、1-、3-配柄附接下不变，还确定了多种接触结构空间及接触同胚群的弱同伦型，部分解决Giroux猜想并证明了Hatcher定理的类似结论。

A remark on the contactomorphism group of overtwisted contact spheres

We show the existence of elements of infinite order in some homotopy groups of the contactomorphism group of overtwisted spheres. It follows in particular that the contactomorphism group of some high dimensional overtwisted spheres is not homotopically equivalent to a finite dimensional Lie group.

On the coarse classification of tight contact structures

We present a sketch of the proof of the following theorems: (1) Every 3-manifold has only finitely many homotopy classes of 2-plane fields which carry tight contact structures. (2) Every closed atoroidal 3-manifold carries finitely many isotopy classes of tight contact structures.

奇数维中的环面可积测地流

Toric integrable geodesic flows in odd dimensions

本文研究奇数维紧致连通黎曼流形上的环面可积测地流，证明在维数不为3的奇数情形或基本群无限时，流形的余球面丛与环面的余球面丛等变接触同胚，从而流形本身同胚于环面。

选择性辛同调及其在接触非压缩现象中的应用

Selective symplectic homology with applications to contact non-squeezing

本文通过构造与刘维尔域边界开子集相关联的选择性辛同调，证明了可由无限维辛同调的刘维尔域填充的同伦球面上存在接触非压缩现象。

Simple homotopy type of the Hamiltonian Floer complex

For an aspherical symplectic manifold, closed or with convex contact boundary, and with vanishing first Chern class, a Floer chain complex is defined for Hamiltonians linear at infinity with coefficients in the group ring of the fundamental group. For two non-degenerate Hamiltonians of the same slope continuation maps are shown to be simple homotopy equivalences. As a corollary the number of contractible Hamiltonian orbits of period 1 can be bounded from below.

Uniqueness and examples of compact toric Sasaki-Einstein metrics

In [11] it was proved that, given a compact toric Sasaki manifold of positive basic first Chern class and trivial first Chern class of the contact bundle, one can find a deformed Sasaki structure on which a Sasaki-Einstein metric exists. In the present paper we first prove the uniqueness of such Einstein metrics on compact toric Sasaki manifolds modulo the action of the identity component of the automorphism group for the transverse holomorphic structure, and secondly remark that the result of [11] implies the existence of compatible Einstein metrics on all compact Sasaki manifolds obtained fr…

Smale收缩的一个接触几何变体

A contact-geometric variant of Smale's contraction

受Smale收缩启发，研究开放旋转对称Darboux球上垂直凸接触形式，证明三维可缩、五维同伦等价于紧支撑微分同胚群，并给出等变乘积分裂。

勒让德缆线空间与塞弗特纤维化链环

Spaces of Legendrian cables and Seifert fibered links

本文确定了标准接触3-球面中具最大瑟斯顿-本内昆不变量的若干勒让德纽结和链环嵌入空间的同伦型，给出充分正缆线的递归公式，证明相关接触结构空间的同伦等价性与可缩性，还发现满足单射h-原理的无穷多新分支。

The existence of contact structures on 9-manifolds

We give necessary and sufficient conditions for a closed orientable 9-manifold M to admit an almost contact structure. The conditions are stated in terms of the Stiefel-Whitney classes of M and other more subtle homotopy invariants of M. By a fundamental result of Borman, Eliashberg and Murphy, M admits an almost contact structure if and only if M admits an over-twisted contact structure. Hence we give necessary and sufficient conditions for M to admit an over-twisted contact structure and we prove that if N is another closed 9-manifold which is homotopy equivalent to M, then M admits an over-…