Generating families and augmentations for Legendrian surfaces

For a Legendrian link Λ ⊂ 𝐽 1 𝑀 with 𝑀 = ℝ or 𝑆 1 , immersed exact Lagrangian fillings 𝐿 ⊂ Symp(𝐽 1 𝑀) ≅ 𝑇 * (ℝ >0 × 𝑀) of Λ can be lifted to conical Legendrian fillings Σ ⊂ 𝐽 1 (ℝ >0 × 𝑀) of Λ. When Σ is embedded, using the version of functoriality for Legendrian contact homology (LCH) from Pan and Rutherford [J. Symplectic Geom. 19 (2021), no. 3, 635-722], for each augmentation 𝛼 ∶ (Σ) → ℤ∕2 of the LCH algebra of Σ, there is an induced augmentation 𝜖 (Σ,𝛼) ∶ (Λ) → ℤ∕2. With Σ fixed, the set of homotopy classes of all such induced augmentations, 𝐼 Σ ⊂ 𝐴𝑢𝑔(Λ)∕∼, is a Legendrian isotopy invariant of Σ. We establish methods to compute 𝐼 Σ based on the correspondence between MCFs and augmentations. This includes developing a functoriality for the cellular differential graded algebra from Rutherford and Sullivan [Adv. Math. 374 (2020), 107348, 71 pp.] 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 ⊂ 𝐽 1 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.M S C 2 0 2 0 53D42 (primary)

show abstract

“…We now review the definition of MCFs for Legendrian knots and surfaces, as in [23,36], allowing for the case of Legendrian cobordisms.…”

Section: Morse Complex Familiesmentioning

confidence: 99%

Augmentations and immersed Lagrangian fillings

Pan

Rutherford

2023

Journal of Topology

Self Cite

View full text Add to dashboard Cite

show abstract

“…In Section 8, for a family of Legendrian twist knots we give examples of augmentations that can be induced in this manner by an immersed filling with 1-double point, but cannot be induced by any oriented embedded filling. In a follow up paper, [26], we undertake a more thorough study of augmentations induced by immersed fillings applying tools from [21,28,29] involving the cellular DGA and the correspondence between Morse complex families and augmentations. In particular, we are able to obtain a flexibility result showing that any (graded) augmentation to Z/2 can be induced by a good Lagrangian filling.…”

Section: Immersed Cobordisms and Augmentationsmentioning

confidence: 99%

Functorial LCH for immersed Lagrangian cobordisms

Pan

Rutherford

2021

Journal of Symplectic Geometry

Self Cite

View full text Add to dashboard Cite

For 1-dimensional Legendrian submanifolds of 1-jet spaces, we extend the functorality of the Legendrian contact homology DGalgebra (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 Σ from Λ − to Λ + , we associate an immersed DGA map, which is a diagramwhere 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. Y. Pan and D. Rutherford 6 The immersed LCH functor I: Construction of immersed maps 687 7 The immersed LCH functor II: Proof of statements 698 8 Augmentations from immersed Lagrangian fillings 706 9 The SFT perspective 714 References 719Functorial LCH for immersed Lagrangian cobordisms 643 Finally in Section 9, we translate the story to Symp(J 1 M ) and show that it matches with the SFT framework. Acknowledgements

show abstract

“…The analogous notion to a Morse complex sequence when dim B = 2 is that of a Morse complex 2-family (MC2F), which was introduced by Rutherford and Sullivan [RS18]. A MC2F is an algebraic gadget which mimics the 2-parametric family of Thom-Smale complexes associated to a generating family.…”

Section: 3mentioning

confidence: 99%

“…Finally, we mention related work of Sullivan and Rutherford [RS18] who used MC2Fs together with their cellular technology for the Chekanov-Eliashberg dga to produce examples of Legendrians in a 1-jet space of a surface B which do not admit generating families on trivial fibre bundles. Their obstruction comes from expressing the action of π 1 (B) on the homology of the fibre H * (F ) in terms of the Chekanov-Eliashberg dga and its augmentations.…”

Section: Dimitroglou-rizell Made the Following Observationmentioning

confidence: 99%