[Paper Review] Sheaves via augmentations of Legendrian surfaces

Dan Rutherford, Michael G. Sullivan|arXiv (Cornell University)|Dec 12, 2019

Homotopy and Cohomology in Algebraic Topology28 references7 citations

TL;DR

This paper constructs cochain complexes of constructible sheaves on $M \times \mathbb{R}$ from augmentations of a new simplicial differential graded algebra (DGA) associated to Legendrian surfaces in $1$-jet spaces. The construction uses combinatorial data from a simplicial decomposition of the base manifold and establishes a map from augmentations to sheaves in the derived category $\mathbf{Sh}^\bullet_\Lambda(M \times \mathbb{R}, \mathbb{K})$, providing a concrete, combinatorial realization of the conjectured link between Legendrian contact homology invariants and microlocal sheaf theory in dimension two.

ABSTRACT

Given an augmentation for a Legendrian surface in a $1$-jet space, $Λ\subset J^1(M)$, we explicitly construct an object, $\mathcal{F} \in Sh_Λ$, of the (derived) category from arXiv:1402.0490 of constructible sheaves on $M imes R$ with singular support determined by $Λ$. In the construction, we introduce a simplicial Legendrian DGA (differential graded algebra) for Legendrian submanifolds in $1$-jet spaces that, based on arXiv:1608.02984 and arXiv:1608.03011, is equivalent to the Legendrian contact homology DGA in the case of Legendrian surfaces. In addition, we extend the approach of arXiv:1402.0490 for $1$-dimensional Legendrian knots to obtain a combinatorial model for sheaves in $Sh_Λ$ in the $2$-dimensional case.

Motivation & Objective

• To establish a concrete, combinatorial construction of sheaves in $\mathbf{Sh}^\bullet_\Lambda(M \times \mathbb{R}, \mathbb{K})$ from augmentations of Legendrian surfaces in $1$-jet spaces.
• To introduce a simplicial DGA over $\mathbb{Z}$-coefficients that generalizes the cellular DGA and agrees with the Legendrian contact homology DGA modulo 2.
• To extend the method of [30] from 1D knots to 2D Legendrian surfaces with mild front singularities, using compatible simplicial decompositions of the base manifold.
• To show that the sheaf construction depends only on the augmentation for fixed underlying sheaf data, making the correspondence explicit and computable.
• To provide a framework for generalizing to higher microlocal rank and non-trivial monodromy via local systems on the Legendrian surface.

Proposed method

• Introduce a simplicial DGA $\mathcal{A}(\Lambda, \mathcal{E})$ for Legendrian surfaces using a compatible simplicial decomposition $\mathcal{E}$ of the base manifold $M$, with coefficients in $\mathbb{Z}$.
• Define chain homotopy diagrams (CHDs) as combinatorial models for augmentations, establishing a bijection between augmentations $\epsilon: \mathcal{A}(\Lambda, \mathcal{E}) \to \mathbb{K}$ and CHDs.
• Construct combinatorial sheaves in $\mathbf{Fun}^\bullet_\Lambda(\mathcal{S}, \mathbb{K})$ by assigning vector spaces to cells and chain maps to incidence relations, using the CHD data.
• Build generalized mapping cylinders from simplex diagrams on edges and vertices, ensuring acyclicity of the total complex via explicit homotopy arguments.
• Use the quasi-equivalence $\Gamma_\mathcal{S}: \mathbf{Sh}^\bullet_\Lambda(\mathcal{S}, \mathbb{K}) \to \mathbf{Fun}^\bullet_\Lambda(\mathcal{S}, \mathbb{K})$ to lift combinatorial sheaves to actual sheaf complexes.
• Verify that the resulting sheaves have singular support contained in the Lagrangian cylinder over $\Lambda$, ensuring they lie in $\mathbf{Sh}^\bullet_\Lambda(M \times \mathbb{R}, \mathbb{K})$.

Experimental results

Research questions

• RQ1 Can augmentations of the Legendrian contact homology DGA for 2-dimensional Legendrian surfaces be used to construct explicit objects in the derived category of constructible sheaves?
• RQ2 Is there a combinatorial, local, and formulaic DGA construction for Legendrian surfaces that generalizes the cellular DGA and works over $\mathbb{Z}$-coefficients?
• RQ3 How can the correspondence between augmentations and sheaves be made concrete and computable in the 2D case, extending the 1D framework of [30]?
• RQ4 What is the role of the simplicial decomposition of the base manifold in organizing the sheaf construction and ensuring compatibility with the DGA structure?
• RQ5 Can the construction be extended to produce sheaves of higher microlocal rank or with non-trivial monodromy via non-trivial local systems on the Legendrian?

Key findings

• The paper constructs a well-defined map $\Phi: \mathit{Aug}(\Lambda; \mathbb{K}) \to \mathbf{Sh}^\bullet_\Lambda(M \times \mathbb{R}, \mathbb{K})$ from augmentations of the simplicial DGA to sheaves in the derived category, establishing a direct link between contact invariants and sheaf theory.
• The simplicial DGA $\mathcal{A}(\Lambda, \mathcal{E})$ is shown to be a DGA over $\mathbb{Z}$, and its mod 2 reduction recovers the cellular DGA, which is known to be stable tame isomorphic to the Legendrian contact homology DGA.
• The construction of sheaves depends only on the augmentation for fixed underlying sheaf data, with differentials determined combinatorially by the augmentation, ensuring a concrete and explicit realization.
• The acyclicity of the total complexes associated to simplex diagrams on edges and vertices is proven via homotopy arguments, ensuring the resulting sheaf complexes are well-defined.
• The resulting sheaves have microlocal rank 1, and the construction can be generalized to higher rank sheaves by replacing $\mathbb{K}$-valued generators with $n$-dimensional vector spaces.
• The authors conjecture that the map from augmentations to sheaves may be essentially surjective onto the subcategory of microlocal rank 1 sheaves with acyclic stalks at infinity, at least over $\mathbb{Z}/2$ and for $M = \mathbb{R}^2$.