Unobstructed Lagrangian cobordism groups of surfaces

TL;DR

This paper computes the Lagrangian cobordism groups of higher genus surfaces, showing they are isomorphic to the Grothendieck group of the derived Fukaya category, using topological and symplectic techniques.

Contribution

It establishes an explicit isomorphism between Lagrangian cobordism groups and the Grothendieck group of the derived Fukaya category for surfaces of genus at least two.

Findings

Cobordism groups are computed and characterized.

Isomorphism with the Grothendieck group of the derived Fukaya category.

Construction of unobstructed Lagrangian cobordisms using topological methods.

Abstract

We study Lagrangian cobordism groups of closed symplectic surfaces of genus $g \geq 2$ whose relations are given by unobstructed, immersed Lagrangian cobordisms. Building upon work of Abouzaid and Perrier, we compute these cobordism groups and show that they are isomorphic to the Grothendieck group of the derived Fukaya category of the surface. The proofs rely on techniques from two-dimensional topology to construct cobordisms that do not bound certain types of holomorphic polygons.