Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors

Abstract

We prove that the wrapped Fukaya category of any $2n$-dimensional Weinstein manifold (or, more generally, Weinstein sector) $W$ is generated by the unstable manifolds of the index $n$ critical points of its Liouville vector field. Our proof is geometric in nature, relying on a surgery formula for Floer cohomology and the fairly simple observation that Floer cohomology vanishes for Lagrangian submanifolds that can be disjoined from the isotropic skeleton of the Weinstein manifold. Note that we do not need any additional assumptions on this skeleton. By applying our generation result to the diagonal in the product $W \times W$, we obtain as a corollary that the open-closed map from the Hochschild homology of the wrapped Fukaya category of $W$ to its symplectic cohomology is an isomorphism, proving a conjecture of Seidel. We work mainly in the "linear setup" for the wrapped Fukaya category, but we also extend the proofs to the "quadratic" and "localisation" setup. This is necessary for dealing with Weinstein sectors and for the applications.