论文标题
增强和沉浸的拉格朗日馅料
Augmentations and immersed Lagrangian fillings
论文作者
论文摘要
对于Legendrian链接$λ\ subset J^1M $，带有$ M = \ Mathbb {r} $或$ s^1 $，沉浸于精确的Lagrangian填充$ legendrian填充$σ\ subset j^1（\ mathbb {r} _ {> 0} \ times m）$ $λ$。当嵌入$σ$时，使用[30]的Legendrian Contact同源性（LCH）的功能版本，对于每个增强$α：\ Mathcal {A}（σ）\ rightArrow \ rightArrow \ rightArrow \ Mathbb {z}/2 $ lch algebra $ $σ$的$ = $ augation $ augation $ augation $ augation $ε \ Mathcal {a}（λ）\ rightarrow \ Mathbb {z}/2 $。使用$σ$固定，所有此类诱导的增强量的同型类别集，$i_σ\ subset \ mathit {ausit {aug}（λ）/{\ sim} $，是$σ$的legendrian同位素不变。我们建立了根据摩尔斯复合族家庭和增强量之间的对应关系计算$i_σ$的方法。这包括从[31]从[31]开发出关于传统的COBORDISM的功能性，并证明了其与LCH功能的等效性。对于任意的$ n \ geq 1 $，我们给出了legendrian torus结的例子，$ 2N $不同的圆锥形legendrian填充物，其诱导的增强套件与众不同。 我们证明，当$ρ\ neq 1 $和$λ\ subset J^1 \ mathbb {r} $每个$ρ$ - 加入$λ$的增强可以通过这种方式通过沉浸式的lagrangian填充来诱导。另外，这被视为对合适的$ρ$ grouned Legendrian Coobordism的合适概念的COBORDISM类计算。
For a Legendrian link $Λ\subset J^1M$ with $M = \mathbb{R}$ or $S^1$, immersed exact Lagrangian fillings $L \subset \mbox{Symp}(J^1M) \cong T^*(\mathbb{R}_{>0} \times M)$ of $Λ$ can be lifted to conical Legendrian fillings $Σ\subset J^1(\mathbb{R}_{>0} \times M)$ of $Λ$. When $Σ$ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from [30], for each augmentation $α: \mathcal{A}(Σ) \rightarrow \mathbb{Z}/2$ of the LCH algebra of $Σ$, there is an induced augmentation $ε_{(Σ,α)}: \mathcal{A}(Λ) \rightarrow \mathbb{Z}/2$. With $Σ$ fixed, the set of homotopy classes of all such induced augmentations, $I_Σ\subset \mathit{Aug}(Λ)/{\sim}$, is a Legendrian isotopy invariant of $Σ$. We establish methods to compute $I_Σ$ based on the correspondence between Morse complex families and augmentations. This includes developing a functoriality for the cellular DGA from [31] with respect to Legendrian cobordisms, and proving its equivalence to the functoriality for LCH. For arbitrary $n \geq 1$, we give examples of Legendrian torus knots with $2n$ distinct conical Legendrian fillings distinguished by their induced augmentation sets. We prove that when $ρ\neq 1$ and $Λ\subset J^1\mathbb{R}$ every $ρ$-graded augmentation of $Λ$ can be induced in this manner by an immersed Lagrangian filling. Alternatively, this is viewed as a computation of cobordism classes for an appropriate notion of $ρ$-graded augmented Legendrian cobordism.
