The mapping class group cannot be realized by homeomorphisms

Let M be a closed surface. By Homeo(M) we denote the group of orientation preserving homeomorphisms of M and let MC(M) denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says that for any closed surface M of genus g ≥ 2, there is no homomorphic section є : MC(M) → Homeo(M) of the standard projection map P : Homeo(M) → MC(M).