Realization of the mapping class group by homeomorphisms

摘要

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 epsilon : MC(M) -> Homeo(M), such that Pr circle epsilon is the identity. This answers a question by Thurston (see [11]).