Special Lagrangian cycles and Calabi-Yau transitions

We construct special Lagrangian 3-spheres in non-K"ahler compact threefolds equipped with the Fu-Li-Yau geometry. These non-K"ahler geometries emerge from topological transitions of compact Calabi-Yau threefolds. From this point of view, a conifold transition exchanges holomorphic 2-cycles for special Lagrangian 3-cycles.