Simplicial descent for Chekanov-Eliashberg dg-algebras

TL;DR

This paper introduces simplicial decompositions for Weinstein manifolds and proves that the Chekanov-Eliashberg dg-algebra satisfies a descent property, enabling explicit computations and connections to Ginzburg dg-algebras.

Contribution

It establishes a descent property for Chekanov-Eliashberg dg-algebras under simplicial decompositions and relates them to Ginzburg dg-algebras for plumbings.

Findings

Chekanov-Eliashberg dg-algebra satisfies descent with respect to simplicial decompositions.

Explicit computation of dg-algebras for plumbings of cotangent bundles.

Chekanov-Eliashberg dg-algebra is quasi-isomorphic to Ginzburg dg-algebra for certain plumbings.

Abstract

We introduce a type of surgery decomposition of Weinstein manifolds 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.