Legendrian contact homology for attaching links in higher dimensional subcritical Weinstein manifolds
Cecilia Karlsson
备注：
27 pages, 9 figures. Comments are welcome! Changes from first version: Focus on subcritical 2n-dimensional Weinstein manifolds with no index n-1-handles. More precise statement of the main theorem. Fixed some minor mistakes and typos
Let $Λ$ be a link of Legendrian spheres in the boundary of a subcritical $2n$-dimensional Weinstein manifold $X$. We show that, under some geometrical assumptions, the computation of the Legendrian contact homology of $Λ$ can be reduced to a computation of Legendrian contact homology in 1--jet spaces. Since the Legendrian contact homology in 1--jet spaces is well studied, this gives a simplified way to compute the Legendrian contact homology of $Λ$. We restrict to the case when the attaching spheres of the subcritical handles of $X$ do not interact with each other, and we assume that there are no handles of index $n-1$. Moreover, we will only consider mod 2 coefficients for now. The more general situation will be addressed in a forthcoming paper. As an application we compute the homology of the free loop space of $\mathbb{CP}^2$.
Let $X$ be a Weinstein manifold with ideal contact boundary $Y$. If $Λ\subset Y$ is a link of Legendrian spheres in $Y$ then by attaching Weinstein handles to $X$ along $Λ$ we get a Weinstein cobordism $X_Λ$ with a collection of Lagrangian co-core disks $C$ corresponding to $Λ$. In \cite{BEE, EL} it was shown that the wrapped Floer cohomology $CW^{\ast}(C)$ of $C$ in the Weinstein manifold $X'_Λ=X\cup X_Λ$is naturally isomorphic to the Legendrian differential graded algebra $CE^{\ast}(Λ)$ of $Λ$ in $Y$. The argument uses properties of moduli spaces of holomorphic curves, the proofs of which we…
The Chekanov-Eliashberg dg-algebra is an algebraic invariant of Legendrian submanifolds of contact manifolds, whose definition recently has been extended to singular Legendrians. We describe a way of constructing simpler models of this dg-algebra for singular Legendrian knots in $\mathbb{R}^{3}$, and give the first examples of singular knots for which the full cohomology can be computed. This includes the $A_{n}$-Legendrians. An important question in the study of the Chekanov-Eliashberg dg-algebra is to understand how its quasi-isomorphism type changes when the Legendrian undergoes Weinstein r…
The Weinstein conjecture in the presence of submanifolds having a Legendrian foliation
Helmut Hessel在1993年引入了一种基于全纯曲线的新技术来证明Weinstein猜想。在这些方法适用的情况中，所有接触3-流形$(M,ξ)$与$π_2(M) \ne 0$。我们修改了Hessen的论点，证明了一些高维接触流形的例子的Weinstein猜想。特别是，我们能够证明与实射影空间的连通和总是具有封闭的可收缩Reeb轨道。
The Chekanov-Eliashberg dg-algebra is a holomorphic curve invariant associated to Legendrian submanifolds of a contact manifold. We extend the definition to Legendrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the new definition gives direct proofs of wrapped Floer cohomology push-out diagrams. It also leads to a proof of a conjectured isomorphism between partially wrapped Floer cohomology and Chekanov-Eliashberg dg-algebras with coefficients in chains on the based loop space.
We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed arboreal Lagrangian skeleta for Weinstein pairs and Weinstein 4-manifolds associated to max-tb Legendrian representatives of algebraic links. We provide computations of Legendrian and Weinstein invariants, and discuss the symplectic topological nature of the Fomin-Pylyavskyy-Shustin-Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems.
We develop the details of a surgery theory for contact manifolds of arbitrary dimension via convex structures, extending the 3-dimensional theory developed by Giroux. The theory is analogous to that of Weinstein manifolds in symplectic geometry, with the key difference that the vector field does not necessarily have positive divergence everywhere. The surgery theory for contact manifolds contains the surgery theory for Weinstein manifolds via a sutured model for attaching critical points of low index. Using this sutured model, we show that the existence of convex structures on closed contact m…
Contact action functional, calculus of variation and canonical generating function of Legendrian submanifolds
Center for Geometry and Physics, Institute for Basic Science (IBS)（基础科学研究所几何物理中心）；POSTECH（浦项科技大学）；CNRS, Laboratoire de Mathématiques Jean Leray, Nantes Université（法国国家科学研究中心让·勒雷数学实验室南特大学）
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