Abstract

The Lagrangian cobordism group is a topological​ invariant of a symplectic manifold which encodes information about the triangulated structure of its Fukaya category (in the sense that it admits a homomorphism onto the Grothendieck group of the latter). In the lowest dimensional case of Riemann surfaces this map is an isomorphism, so the cobordism group `sees all of the triangulated structure’; whether this is the case in higher dimensions is an open problem. Recent results of Sheridan-Smith show that Lagrangian cobordism groups of symplectic 4-manifolds with trivial canonical bundle are so large and complicated as to make a direct computation unfeasible. In this talk I will consider a symplectic 4-manifold whose canonical bundle is torsion but not trivial, and explain that (a certain subgroup of) the cobordism group can be directly computed in this case. Then, using homologicla mirror symmetry and the computation of the Chow groups of the mirror variety, I will show that this subgroup maps isomorphically onto the Grothendieck group of the Fukaya category.