A geometric criterion for generating the Fukaya category

Mohammed Abouzaid

Published 2010-01-26, updated 2010-09-29Version 3

Given a collection of exact Lagrangians in a Liouville manifold, we construct a map from the Hochschild homology of the Fukaya category that they generate to symplectic cohomology. Whenever the identity in symplectic cohomology lies in the image of this map, we conclude that every Lagrangian lies in the idempotent closure of the chosen collection. The main new ingredients are (1) the construction of operations controlled by discs with two outputs on the Fukaya category, and (2) the Cardy relation.