2020/06/29 by Yu Pan, Pan, Yu, Dan Rutherford +1 · 2 citations
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Algebraic structures and combinatorial models
paper · pdf · doi:10.48550/arxiv.2006.16436
For a Legendrian link Λ⊂ J1M with M = ℝ or S1, immersed exact Lagrangian fillings L ⊂ Symp(J1M) ≅ T^*(ℝ>0 × M) of Λ can be lifted to conical Legendrian fillings Σ⊂ J1(ℝ>0 × M) of Λ. When Σ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from [30], for each augmentation α: A(Σ) → ℤ/2 of the LCH algebra of Σ, there is an induced augmentation ε(Σ,α): A(Λ) → ℤ/2. With Σ fixed, the set of homotopy classes of all such induced augmentations, IΣ⊂ Aug(Λ)/∼, is a Legendrian isotopy invariant of Σ. We establish methods to compute IΣ 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 ≥ 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≠ 1 and Λ⊂ J1ℝ 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.