Symplectic hats

Non-Exact Symplectic Cobordisms Between Contact 3-Manifolds

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to the tight 3-sphere, or which admit symplectic caps containing symplectically embedded spheres with vanishing self-intersection. These constructions imply new and simplified proofs of several recent results involving fillability, planarity and non-separating contact type embeddin…

An invariant of link cobordisms from symplectic Khovanov homology

Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fixed pair of links. These morphisms define a functor from the category of links and such cobordisms to the category of abelian groups and group homomorphisms up to a sign ambiguity. This provides an extra structure for symplectic Khovanov homology and more generally an isotopy inva…

Open books and exact symplectic cobordisms

Given two open books with equal pages we show the existence of an exact symplectic cobordism whose negative end equals the disjoint union of the contact manifolds associated to the given open books, and whose positive end induces the contact manifold associated to the open book with the same page and concatenated monodromy. Using similar methods we show the existence of strong fillings for contact manifolds associated with doubled open books, a certain class of fiber bundles over the circle obtained by performing the binding sum of two open books with equal pages and inverse monodromies. From …

Legendrian knots and exact Lagrangian cobordisms

We introduce constructions of exact Lagrangian cobordisms with cylindrical Legendrian ends and study their invariants which arise from Symplectic Field Theory. A pair $(X,L)$ consisting of an exact symplectic manifold $X$ and an exact Lagrangian cobordism $L\subset X$ which agrees with cylinders over Legendrian links $Λ_+$ and $Λ_-$ at the positive and negative ends induces a differential graded algebra (DGA) map from the Legendrian contact homology DGA of $Λ_+$ to that of $Λ_-$. We give a gradient flow tree description of the DGA maps for certain pairs $(X,L)$, which in turn yields a purely c…

Naturality of the Contact Invariant in Monopole Floer Homology Under Strong Symplectic Cobordisms

The contact invariant is an element in the monopole Floer homology groups of an oriented closed three manifold canonically associated to a given contact structure. A non-vanishing contact invariant implies that the original contact structure is tight, so understanding its behavior under symplectic cobordisms is of interest if one wants to further exploit this property. By extending the gluing argument of Mrowka and Rollin to the case of a manifold with a cylindrical end, we will show that the contact invariant behaves naturally under a strong symplectic cobordism. As quick applications of the …

Orbifold adjunction formula and symplectic cobordisms between lens spaces

Each lens space has a canonical contact structure which lifts to the distribution of complex lines on the three-sphere. In this paper, we show that a symplectic homology cobordism between two lens spaces, which is given with the canonical contact structure on the boundary, must be diffeomorphic to the product of a lens space with the unit interval. As one of the main ingredients in the proof, we also derive in this paper the adjunction and intersection formulae for pseudoholomorphic curves in an almost complex 4-orbifold, extending the relevant work of Gromov and McDuff in the manifold setting…

Cobordism maps on periodic Floer homology induced by elementary Lefschetz fibrations

Periodic Floer homology (PFH) is a Gromov-Floer type invariant for fibered three-manifolds with Hamiltonian structures. The cobordism maps on periodic Floer homology induced by symplectic cobordisms are currently only defined indirectly by using Seiberg-Witten theory. In this paper, we investigate the cobordism maps induced by a class of symplectic cobordisms constructed by P. Seidel, called elementary Lefschetz brations. We define the PFH cobordism maps induced by elementary Lefschetz fibrations in terms of holomorphic curves. Moreover, we compute these maps for some cases.

Smooth s-cobordisms of elliptic 3-manifolds

The main result of this paper states that a symplectic s-cobordism of elliptic 3-manifolds is diffeomorphic to a product (assuming a canonical contact structure on the boundary). Based on this theorem, we conjecture that a smooth s-cobordism of elliptic 3-manifolds is smoothly a product if its universal cover is smoothly a product. We explain how the conjecture fits naturally into the program of Taubes of constructing symplectic structures on an oriented smooth 4-manifold from generic self-dual harmonic forms. The paper also contains an auxiliary result of independent interest, which generaliz…

辛流形的单位余切丛的填充及其应用

Symplectic fillings of unit cotangent bundles of spheres and applications

该研究证明了奇数维球面的单位余切丛的辛填充在微分同胚意义下唯一，并探讨了其在辛cobordism和闭合辛流形中的应用。

Constructions of Lagrangian cobordisms

Lagrangian cobordisms between Legendrian knots arise in Symplectic Field Theory and impose an interesting and not well-understood relation on Legendrian knots. There are some known "elementary" building blocks for Lagrangian cobordisms that are smoothly the attachment of $0$- and $1$-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordism can be related by a ribbon Lagrangian cobordism, in particular one that is "decomposable" into a composition of these elementary building blocks. We will describe these and other combi…

Lagrangian Cobordism and Fukaya Categories

Given a symplectic manifold M, we consider a category with objects finite ordered families of Lagrangian submanifolds of M (subject to certain additional constraints) and with morphisms Lagrangian cobordisms relating them. We construct a functor that maps this category to a variant of the derived Fukaya category of M in a way that takes into account the triangulated structure of the latter.

Mazur 流形与辛结构

Mazur manifolds and symplectic structures

利用Heegaard Floer同调的cobordism映射，阻碍了Akbulut-Kirby流形上存在辛结构的可能性，并表明存在具有确定交形式的exotic对。

A transverse knot invariant from Z2-equivariant Heegaard Floer cohomology

We define an invariant of based transverse links, as a well-defined element inside the equivariant Heegaard Floer cohomology of its branched double cover, defined by Lipschitz, Hendricks, and Sarkar. We prove the naturality and functoriality of equivariant Heegaard Floer cohomology for branched double covers of $S^3$ along based knots, and then prove that our transverse link invariant $c_{\mathbb{Z}_{2}}(ξ_{K})$ is an well-defined element which is always nonvanishing and functorial under certain classes of symplectic cobordisms, and describe its behavior under negative stabilization. It follow…

环面纽结Reeb动力学与Calabi不变量

Torus knotted Reeb dynamics and the Calabi invariant

本文针对标准紧三维球面上允许标准横向正T(p,q)环面纽结为椭圆Reeb轨道的接触形式，建立次级Reeb轨道集的存在性，推导相关保面积微分同胚周期轨道平均作用量上界，发展纽结过滤嵌入接触同调并得到精确辛余维存在障碍。