Simplicial descent for Chekanov–Eliashberg dg‐algebras

Abstract

AbstractWe introduce a type of surgery decomposition of Weinstein manifolds that we call simplicial decompositions. The main result of this paper is that the Chekanov–Eliashberg dg‐algebra of the attaching spheres of a Weinstein manifold satisfies a descent (cosheaf) property with respect to a simplicial decomposition. Simplicial decompositions generalize the notion of Weinstein connected sum and we show that there is a one‐to‐one correspondence (up to Weinstein homotopy) between simplicial decompositions and so‐called good sectorial covers. As an application, we explicitly compute the Chekanov–Eliashberg dg‐algebra of the Legendrian attaching spheres of a plumbing of copies of cotangent bundles of spheres of dimension at least three according to any plumbing quiver. We show by explicit computation that this Chekanov–Eliashberg dg‐algebra is quasi‐isomorphic to the Ginzburg dg‐algebra of the plumbing quiver.