Simplicial descent for Chekanov-Eliashberg dg-algebras

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. Contents 19 3.1. Simplicial decompositions and good sectorial covers 19 3.2. Relation to sectorial descent 26 4. Applications 27 4.1. Legendrian submanifolds-with-boundary-and-corners 27 4.2. The Chekanov-Eliashberg cosheaf 30 4.3. Plumbing of cotangent bundles of spheres 31 References 36

Tangle contact homology

“…In this section we generalize the notion of a simplicial decomposition of a Weinstein manifold introduced in [Asp21b] to Weinstein pairs. We assume in the following that the reader is familiar with the construction of a simplicial decomposition of a Weinstein manifold as defined in [Asp21b, Section 2.2].…”

“…Recall that a Weinstein pair (X 2n , P 2n−2 ) corresponds to a Weinstein sector X via convex completion [GPS20, Section 2.7]. We may thus think about constructions in this section as a generalization of the simplicial decomposition of a Weinstein manifold as constructed in [Asp21b] to a simplicial decomposition of a Weinstein sector.…”

“…(1) It follows from the construction of simplicial handles from [Asp21b] that Reeb chords of ∆ P (h ) ⊂ ∂X 0 of action < a (possibly after shrinking the size of the simplicial handles) are contained in π −1 (0, 0) where π :…”

“…Method of proof. The method used to prove Theorem 1.1 is a generalization of the gluing formulas proven in [AE21,Asp21b], which we now give a brief sketch of. First considering two Weinstein pairs (X 1 , V ) and (X 2 , V ), we can glue X 1 and X 2 together along their common Weinstein hypersurface V 2n−2 → ∂X 2n to obtain a new Weinstein manifold denoted by X 1 # V X 2 .…”

“…First considering two Weinstein pairs (X 1 , V ) and (X 2 , V ), we can glue X 1 and X 2 together along their common Weinstein hypersurface V 2n−2 → ∂X 2n to obtain a new Weinstein manifold denoted by X 1 # V X 2 . This operation is called Weinstein connected sum [Avd21,Eli18,ÁGEN20], and gives gluing formulas for the Chekanov-Eliashberg dg-algebra with field coefficients of the Legendrian attaching link [AE21,Asp21b].…”