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Realization of the mapping class group by homeomorphisms

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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]).

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Authors and Affiliations

  1. Institute of Mathematics, University of Warwick, Coventry, CV4 7AL, UK

    Vladimir Markovic

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  1. Vladimir Markovic
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Correspondence to Vladimir Markovic.

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Mathematics Subject Classification (2000)

Primary 20H10, 37F30

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Markovic, V. Realization of the mapping class group by homeomorphisms. Invent. math. 168, 523–566 (2007). https://doi.org/10.1007/s00222-007-0039-0

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