Holomorphic curves for Legendrian surgery

Abstract

Let $X$ be a Weinstein manifold with ideal contact boundary $Y$. If $\Lambda\subset Y$ is a link of Legendrian spheres in $Y$ then by attaching Weinstein handles to $X$ along $\Lambda$ we get a Weinstein cobordism $X_{\Lambda}$ with a collection of Lagrangian co-core disks $C$ corresponding to $\Lambda$. In \cite{BEE, EL} it was shown that the wrapped Floer cohomology $CW^{\ast}(C)$ of $C$ in the Weinstein manifold $X'_{\Lambda}=X\cup X_{\Lambda}$is naturally isomorphic to the Legendrian differential graded algebra $CE^{\ast}(\Lambda)$ of $\Lambda$ in $Y$. The argument uses properties of moduli spaces of holomorphic curves, the proofs of which were only sketched. The purpose of this paper is to provide proofs of these properties.