On the Fukaya Categories of Higher Genus Surfaces

We construct the Fukaya category of a closed surface equipped with an area form using only elementary (essentially combinatorial) methods. We also compute the Grothendieck group of its derived category.Comment: 40 pages, 17 figures. Final Versio

The family Floer functor is faithful

Family Floer theory yields a functor from the Fukaya category of a symplectic manifold admitting a Lagrangian torus fibration to a (twisted) category of perfect complexes on the mirror rigid analytic space. This functor is shown to be faithful by a degeneration argument involving moduli spaces of annuli.Comment: 70 pages, 24 figures. Final version, with substantially enhanced exposition, accepted for publication at JEM

A geometric criterion for generating the Fukaya category

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.Comment: 42 pages, 4 figures. Minor changes. Final version to appear in Publ. IHE

A cotangent fibre generates the Fukaya category

We prove that the algebra of chains on the based loop space recovers the derived (wrapped) Fukaya category of the cotangent bundle of a closed smooth orientable manifold. The main new idea is the proof that a cotangent fibre generates the Fukaya category using a version of the map from symplectic cohomology to the homology of the free loop space introduced by Cieliebak and Latschev.Comment: 40 pages, 10 figures. Minor changes. Final version to appear in Advances in Mathematic