Realization of the mapping class group by homeomorphisms

Abstract

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 E:MC(M)→Homeo(M), such that Pr∘E is the identity. This answers a question by Thurston (see [11]).

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