Generating families and constructible sheaves

2015/04/06 by Vivek Shende, Shende, Vivek
Mathematics · Medicine · #Botulinum Toxin and Related Neurological Disorders #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1504.01336

openalex publication_date 2015/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Λ be a Legendrian in the jet space of some manifold X. To a generating family presentation of Λ, we associate a constructible sheaf on X × ℝ whose singular support at infinity is Λ, and such that the generating family homology is canonically isomorphic to the endomorphism algebra of this sheaf. That is, the theory of generating family homology embeds in sheaf theory, and more specifically in the category studied in [STZ]. When X = ℝ, i.e., for the theory of Legendrian knots and links in the standard contact ℝ³, we use ideas from the proof of the h-cobordism theorem to show this embedding is an equivalence. Combined with the results of [NRSSZ], this implies in particular that the generating family homologies of a knot are the same as its linearized Legendrian contact homologies.

Citations

• Quantization of conic Lagrangian submanifolds of cotangent bundles
• Generalization of the Chekanov theorem: diameters of immersed manifolds and wave fronts

Related

• Sheaves via augmentations of Legendrian surfaces
• A product structure on Generating Family Cohomology for Legendrian Submanifolds
• A stable homotopy invariant for Legendrians with generating families
• Families of Legendrian Submanifolds via Generating Families
• Augmentations and Rulings of Legendrian Knots