From 3D topological quantum field theories to 4D models with defects

(2+1) dimensional topological quantum field theories with defect excitations are by now quite well understood, while many questions are still open for (3+1) dimensional TQFTs. Here we propose a strategy to lift states and operators of a (2+1) dimensional TQFT to states and operators of a (3+1) dimensional theory with defects. The main technical tool are Heegard splittings, which allow to encode the topology of a three--dimensional manifold with line defects into a two--dimensional Heegard surface. We apply this idea to the example of BF theory which describes locally flat connections. This shows in particular how the curvature excitation generating surface operators of the (3+1) dimensional theory can be obtained from closed ribbon operators of the (2+1) dimensional BF theory. We hope that this technique allows the construction and study of more general models based on unitary fusion categories.