Microlocal Category for Weinstein Manifolds via the h-Principle

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 with 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. The invariance of microlocal sheaves along such an 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.