Fukaya category of surfaces and mapping class group action

TL;DR

This paper constructs the Fukaya category for higher genus surfaces, computes its Grothendieck group, and demonstrates the faithful action of the surface's mapping class group on this category, advancing understanding in symplectic topology.

Contribution

It introduces a topological variant of the Fukaya category for surfaces, disregarding area form and using an admissibility condition, and proves the faithfulness of the mapping class group action.

Findings

Constructed the Fukaya category for surfaces of genus > 1.

Computed the Grothendieck group of this category.

Established the faithfulness of the mapping class group action.

Abstract

We construct the Fukaya category of a surface with genus greater than one and compute its Grothendieck group. We consider here a topological variant, in which we disregard the area form and use instead an admissibility condition borrowed from Heegaard-Floer theory which ensures invariance under isotopy. We also study action of the mapping class group of the surface on its Fukaya category and establish faithfulness.