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Lagrangian cobordism functor in microlocal sheaf theory I

Wenyuan Li

TL;DR

The paper constructs a Lagrangian cobordism functor in microlocal sheaf theory, producing a map between compact-object sheaf categories supported on Λ_+ and Λ_- together with a right adjoint on proper objects, formulated via Nadler–Shende’s framework. The approach embeds Weinstein skeleta into larger contact boundaries and uses nearby cycles to define Φ_L^*, with concatenations of cobordisms corresponding to functor compositions; the construction is shown to be fully faithful and to admit adjoints, independently of Floer theory. It establishes base-change, Hamiltonian-invariance, and comparisons to existing sheaf-quantization and filling formalisms, yielding exact triangles and Mayer–Vietoris-type obstructions. The paper also applies these results to Legendrian surfaces, deriving obstructions to cobordisms among complex families of Legendrian weaves and connecting to moduli of microlocal rank-1 sheaves. Altogether, it provides a purely sheaf-theoretic, functorial description of Lagrangian cobordisms with concrete implications for Legendrian topology and obstructions to cobordisms.

Abstract

Given a Lagrangian cobordism $L$ of Legendrian submanifolds from $Λ_-$ to $Λ_+$, we construct a functor $Φ_L^*: Sh^c_{Λ_+}(M) \rightarrow Sh^c_{Λ_-}(M) \otimes_{C_{-*}(Ω_*Λ_-)} C_{-*}(Ω_*L)$ between sheaf categories of compact objects with singular support on $Λ_\pm$ and its right adjoint on sheaf categories of proper objects, using Nadler-Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds.

Lagrangian cobordism functor in microlocal sheaf theory I

TL;DR

The paper constructs a Lagrangian cobordism functor in microlocal sheaf theory, producing a map between compact-object sheaf categories supported on Λ_+ and Λ_- together with a right adjoint on proper objects, formulated via Nadler–Shende’s framework. The approach embeds Weinstein skeleta into larger contact boundaries and uses nearby cycles to define Φ_L^*, with concatenations of cobordisms corresponding to functor compositions; the construction is shown to be fully faithful and to admit adjoints, independently of Floer theory. It establishes base-change, Hamiltonian-invariance, and comparisons to existing sheaf-quantization and filling formalisms, yielding exact triangles and Mayer–Vietoris-type obstructions. The paper also applies these results to Legendrian surfaces, deriving obstructions to cobordisms among complex families of Legendrian weaves and connecting to moduli of microlocal rank-1 sheaves. Altogether, it provides a purely sheaf-theoretic, functorial description of Lagrangian cobordisms with concrete implications for Legendrian topology and obstructions to cobordisms.

Abstract

Given a Lagrangian cobordism of Legendrian submanifolds from to , we construct a functor between sheaf categories of compact objects with singular support on and its right adjoint on sheaf categories of proper objects, using Nadler-Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high dimensional Legendrian submanifolds.

Paper Structure

This paper contains 30 sections, 34 theorems, 216 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Main Results
  4. Relations with Other Works
  5. Relation with sheaf quantization of Legendrian isotopy
  6. Relation with sheaf quantization of Lagrangian fillings
  7. Other proposals of the cobordism functor
  8. Applications to Legendrian Surfaces
  9. Organization of the Paper
  10. Conventions
  11. Preliminaries in Sheaf Theory
  12. Singular Supports
  13. Microlocalization and $\mu Sh$
  14. Functors for Hamiltonian Isotopies
  15. Sheaf Quantization of Weinstein Manifolds
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $X$ be a Weinstein manifold with subanalytic skeleton $\mathfrak{c}_X$, $\Lambda_-, \Lambda_+ \subset \partial_\infty X$ be Legendrian submanifolds, and $L \subset \partial_\infty X \times \mathbb{R}$ an exact Lagrangian cobordism from $\Lambda_-$ to $\Lambda_+$. There is a cobordism functor bet and a fully faithful adjoint functor between microlocal sheaf categories of proper objects such th

Figures (13)

  • Figure 1: The front projection $\pi_\text{front}: J^1(\Sigma) \rightarrow \Sigma \times \mathbb{R}$ of the Legendrian surfaces corresponding to each planar $N$-graph on the top.
  • Figure 2: On the left is the Clifford Legendrian torus and its corresponding 2-graph, and on the right is the unknotted Legendrian torus and its corresponding 2-graph.
  • Figure 3: The singular support of a sheaf and the combinatoric description.
  • Figure 4: The diagram of maps in the proof of Theorem \ref{['concate']}.
  • Figure 5: The Nadler-Shende construction (left) and the Jin-Treumann construction (right). The grey regions are the supports of the corresponding sheaves. The thin lines on the left are the skeleton $M \cup \nu^*_{U,+}M$ embedded in $J^1(M)$, and the thick lines there are the two copies of Lagrangian fillings. The blue lines are the family of cusps $\partial \Lambda \,\times \prec$.
  • ...and 8 more figures

Theorems & Definitions (98)

  • Definition 1.1
  • Remark 1.1
  • Definition 1.2
  • Remark 1.2
  • Theorem 1.1
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Corollary 1.2
  • ...and 88 more