For $1$-dimensional Legendrian submanifolds of $1$-jet spaces, we extend the functorality of the Legendrian contact homology DG-algebra (DGA) from embedded exact Lagrangian cobordisms, as in [16], to a class of immersed exact Lagrangian cobordisms by considering their Legendrian lifts as conical Legendrian cobordisms. To a conical Legendrian cobordism $\Sigma$ from $\Lambda_{-}$ to $\Lambda_{+}$, we associate an immersed DGA map, which is a diagram\[\mathcal{A}(\Lambda_{+}) \overset{f}{\to} \mathcal{A}(\Sigma) \overset{i}{\hookleftarrow} \mathcal{A}(\Lambda_{-}) \quad \textrm{,}\]where $f$ is a DGA map and $i$ is an inclusion map. This construction gives a functor between suitably defined categories of Legendrians with immersed Lagrangian cobordisms and DGAs with immersed DGA maps. In an algebraic preliminary, we consider an analog of the mapping cylinder construction in the setting of DG-algebras and establish several of its properties. As an application we give examples of augmentations of Legendrian twist knots that can be induced by an immersed filling with a single double point but cannot be induced by any orientable embedded filling.

中文翻译：

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浸入式拉格朗日协边函数的函子 LCH

对于 $1$-jet 空间的 $1$-维 Legendrian 子流形，我们将 Legendrian 接触同调 DG-代数 (DGA) 的函式从嵌入的精确拉格朗日协边扩展到一类浸入式精确拉格朗日协边，如[16]将他们的 Legendrian 升降机视为锥形 Legendrian cobordisms。对于从 $\Lambda_{-}$ 到 $\Lambda_{+}$ 的圆锥形勒让德坐标 $\Sigma$，我们关联了一个浸入式 DGA 映射，它是一个图\[\mathcal{A}(\Lambda_{+} ) \overset{f}{\to} \mathcal{A}(\Sigma) \overset{i}{\hookleftarrow} \mathcal{A}(\Lambda_{-}) \quad \textrm{,}\]where $f$ 是 DGA 映射，$i$ 是包含映射。这种构造在适当定义的具有浸入式拉格朗日坐标系的勒格朗日函数和具有浸入式 DGA 映射的 DGA 之间给出了一个函子。在代数初步中，我们在 DG 代数的设置中考虑映射圆柱结构的模拟，并建立它的几个属性。作为一个应用，我们给出了勒让德扭结增强的例子，这种增强可以由具有单个双点的浸入填充物引起，但不能由任何可定向的嵌入填充物引起。