Realization of the mapping class group by homeomorphisms

In this paper, we show that the mapping class group of a closed surface can not be geometrically realized as a group of homeomorphisms of that surface. More precisely, let Pr: Homeo(M) → MC(M) denote the standard projection of the group of homeomorphisms to the mapping class group of a closed surface M of genus g>5. We show that there is no homomorphism ε: MC(M) → Homeo(M), such that Pr o ε is the identity. This answers a question by Thurston (see [11]). © Springer-Verlag 2007.