It is well known that analytic invariants totally determine the topology of complex curves. Hodge theory gives a process to obtain topological information on Kähler manifolds from the the spaces of holomorphic differential forms. We in this talk analyse if there are topological implications to be derived from the spaces of symmetric differentials on complex surfaces. The dimensions of the spaces of symmetric differentials do not encode topological information, but we will show that there is a special type of symmetric differentials, which we call closed, which reflect topological properties.