Inventiones mathematicae

, Volume 168, Issue 3, pp 523–566

Realization of the mapping class group by homeomorphisms


DOI: 10.1007/s00222-007-0039-0

Cite this article as:
Markovic, V. Invent. math. (2007) 168: 523. doi:10.1007/s00222-007-0039-0


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 \(Pr:\mathcal{H}\textit{omeo}(M)\to\mathcal{M}\mathcal{C}(M)\) 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 \(\mathcal{E}:\mathcal{M}\mathcal{C}(M)\to\mathcal{H}\textit{omeo}(M)\), such that \(Pr\circ\mathcal{E}\) is the identity. This answers a question by Thurston (see [11]).

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of WarwickCoventryUK