The mapping class group cannot be realized by homeomorphisms

Dragomir Saric, CUNY Queens

Let M be a closed surface. Let Homeo(M) be the group of orientation preserving homeomorphisms of M and let MC(M) be the Mapping class group. Markovic proved the conjecture of Thurston that there is no homomorphic section E:MC(M) \to Homeo(M) of the standard projection map Proj:Homeo(M) \to MC(M) when the genus of M is at least six. We complete the proof of the conjecture of Thurston for closed surfaces of genus at least two. Our methods give a unified proof for any genus. Joint work with V. Markovic.