Lagrangian cobordism functor in microlocal sheaf theory I

Abstract

Given a Lagrangian cobordism $L$ of Legendrian submanifolds from ${\Lambda }_{-}$ to ${\Lambda }_{+}$, we construct a functor ${\Phi }_L^{\ast }:S{h}_{{\Lambda }_{+}}^c(M)\to S{h}_{{\Lambda }_{-}}^c(M){\otimes }_{C_{-\ast }({\Omega }_{\ast }{\Lambda }_{-})}{C}_{-\ast }({\Omega }_{\ast }L)$ between sheaf categories of compact objects with singular support on ${\Lambda }_{\pm }$ 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.

arXiv comment43 pages, 13 figures. v2: Minor revisions on Intro, Sec 2.2 and other places (theorem numbers changed in Sec 3). v3: Fixed the proof of Thm 2.10 and Prop 3.3 (Lem 3.4 in the previous version) based on referees' comments, revisions on Sec 2.4, 4.1 and other places (section and theorem numbers changed in Sec 3; title changed). To appear in J Topol. v4, v5: Minor modification: Thm 1.9 & Rem 2.13

References in corpus (20)

• Loose Legendrian embeddings in high dimensional contact manifolds
• Duality between Lagrangian and Legendrian invariants
• Wrapped microlocal sheaves on pairs of pants
• Microlocal Morse theory of wrapped Fukaya categories
• Non-characteristic expansions of Legendrian singularities
• Floer theory for Lagrangian cobordisms
• Quantization of conic Lagrangian submanifolds of cotangent bundles
• The conormal torus is a complete knot invariant
• Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
• The contact homology of Legendrian knots with maximal Thurston-Bennequin invariant
• Holomorphic curves for Legendrian surgery
• Augmentations, Fillings, and Clusters
• Sheaf quantization in Weinstein symplectic manifolds
• Orlov and Viterbo functors in partially wrapped Fukaya categories
• Simplifying Weinstein Morse functions
• Sheaves via augmentations of Legendrian surfaces
• Microlocal sheaf categories and the $J$-homomorphism
• Sheaf Quantization of Legendrian Isotopy
• Geometric and algebraic presentations of Weinstein domains
• Functorial LCH for immersed Lagrangian cobordisms