Fukaya categories of higher-genus surfaces and pants decompositions

Abstract

In this paper we prove a local-to-global principle for the Fukaya category of a closed Riemann surface $\Sigma$ of genus $g \geq 2$. We show that $\mathrm{Fuk}(\Sigma)$ can be glued from the Fukaya categories of the pairs-of-pants making up a pants decomposition of $\Sigma$. This extends our earlier results for the case of punctured Riemann surfaces. Our result has several interesting consequences: we obtain simple proofs of old and new HMS statements for Riemann surfaces, and establish a geometrization theorem for the objects of $\mathrm{Fuk}(\Sigma)$.