Lagrangian cobordisms and K-theory of symplectic bielliptic surfaces

We consider a family of closed symplectic manifolds 4-manifolds which we call symplectic bielliptic surfaces and study its Lagrangian cobordism group of weakly-exact Lagrangian G-branes (that is, Lagrangians equipped with a grading, a Pin structure and a G-local system); relations come from Lagrangian cobordisms satisfying a tautologically unobstructedness-type condition, also equipped with G-brane structures. Our first theorem computes its subgroup generated by tropical Lagrangians. When G is the unitary group of the Novikov field, we use homological mirror symmetry to compute the Grothendieck group of the Fukaya category and show it agrees with our computation for the cobordism group. This leads us to conjecture that tropical Lagrangians generate the whole cobordism group.