Microlocal category for Weinstein manifolds via h-principle

TL;DR

This paper constructs a microlocal categorical invariant for Weinstein manifolds using the h-principle, providing a new perspective on their symplectic topology through sheaf theory.

Contribution

It introduces a constructible co/sheaf of categories on the skeleton of Weinstein manifolds, relying only on homotopy classes and the h-principle, and establishes its invariance properties.

Findings

Defines a constructible co/sheaf of categories on Weinstein skeletons.

Uses the h-principle to guarantee existence and isotopy invariance of embeddings.

Proposes invariance of global sections under Liouville deformation, pending proof.

Abstract

On a Weinstein manifold, we define a constructible co/sheaf of categories on the skeleton. The construction works with arbitrary coefficients, and depends only on the homotopy class of a section of the Lagrangian Grassmannian of the stable symplectic normal bundle. The definition is as follows. Take any, possibly high codimension, exact embedding into a cosphere bundle. Thicken to a hypersurface, and consider the Kashiwara-Schapira stack along the thickened skeleton. Pull back along the inclusion of the original skeleton. Gromov's h-principle for contact embeddings guarantees existence and uniqueness up to isotopy of such an embedding. Invariance of microlocal sheaves along such isotopy is well known. We expect, but do not prove here, invariance of the global sections of this co/sheaf of categories under Liouville deformation.