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On Lagrangian cobordisms and the Chekanov-Eliashberg DGA

Sierra Knavel, Thomas Rodewald

TL;DR

The paper develops a framework to understand how exact Lagrangian cobordisms influence Legendrian invariants, focusing on the Chekanov-Eliashberg DGA and its linearization. It defines the augmented cobordism map $\Phi_L^{\epsilon_1}$ and proves it yields a chain map on the linearized complex, with the trivial cylinder recovering the identity and isotopy-invariance under mild hypotheses. Furthermore, the work lifts to an $A_\infty$-morphism between linearized cochain algebras, showing that higher-order products are preserved under cobordism maps, up to suitable conditions. This provides a robust algebraic mechanism to compare Legendrian ends across cobordisms and informs how higher invariants behave under Lagrangian deformations.

Abstract

In this paper, we consider exact Lagrangian cobordisms and the map they induce on the Chekanov-Eliashberg DGAs of their Legendrian ends as defined by Ekholm, Honda, and Kalman. Specifically, we show how to adapt this map to linearizations of the DGA using augmentations. We then show its induced map on linearized Legendrian contact homology is invariant under Lagrangian isotopy under mild hypotheses, as well as its induced map on higher order product structures.

On Lagrangian cobordisms and the Chekanov-Eliashberg DGA

TL;DR

The paper develops a framework to understand how exact Lagrangian cobordisms influence Legendrian invariants, focusing on the Chekanov-Eliashberg DGA and its linearization. It defines the augmented cobordism map and proves it yields a chain map on the linearized complex, with the trivial cylinder recovering the identity and isotopy-invariance under mild hypotheses. Furthermore, the work lifts to an -morphism between linearized cochain algebras, showing that higher-order products are preserved under cobordism maps, up to suitable conditions. This provides a robust algebraic mechanism to compare Legendrian ends across cobordisms and informs how higher invariants behave under Lagrangian deformations.

Abstract

In this paper, we consider exact Lagrangian cobordisms and the map they induce on the Chekanov-Eliashberg DGAs of their Legendrian ends as defined by Ekholm, Honda, and Kalman. Specifically, we show how to adapt this map to linearizations of the DGA using augmentations. We then show its induced map on linearized Legendrian contact homology is invariant under Lagrangian isotopy under mild hypotheses, as well as its induced map on higher order product structures.
Paper Structure (8 sections, 14 theorems, 68 equations, 2 figures)

This paper contains 8 sections, 14 theorems, 68 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Preliminaries
  4. Chekanov-Eliashberg DGA
  5. Chekanov-Eliashberg DGA cobordism map
  6. The $A_\infty$ structure
  7. Linearizing the cobordism map
  8. Preserving the $A_\infty$ structure

Key Result

Proposition 1.1

Let $L$ be an exact Lagrangian cobordism from $\Lambda_1$ to $\Lambda_2$, and $\epsilon_1$ an augmentation of $(\mathcal{A}_{\Lambda_1}, \partial_{\Lambda_1})$ and $\epsilon_2 = \epsilon_1 \circ \Phi_L$. Then

Figures (2)

  • Figure 1: The front projection of the max-tb unknot, left, and its Lagrangian projection, right.
  • Figure 2: A schematic of a Lagrangian cobordism from a Legendrian unknot to a Legendrian trefoil in the symplectization of $(\mathbb{R}^3, \xi_{std})$.

Theorems & Definitions (25)

  • Proposition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • proof
  • ...and 15 more