, Volume 168, Issue 3, pp 523-566
Date: 07 Mar 2007

Realization of the mapping class group by homeomorphisms

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access

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 \(Pr:\mathcal{H}\textit{omeo}(M)\to\mathcal{M}\mathcal{C}(M)\) 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]).

Mathematics Subject Classification (2000)

Primary 20H10, 37F30