Commentarii Mathematici Helvetici

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Volume 88, Issue 1, 2013, pp. 205–220
DOI: 10.4171/CMH/283

Published online: 2013-01-07

Some groups of mapping classes not realized by diffeomorphisms

Mladen Bestvina[1], Thomas Church[2] and Juan Souto[3]

(1) University of Utah, Salt Lake City, USA
(2) Stanford University, USA
(3) Université de Rennes 1, France

Let $\Sigma$ be a closed surface of genus $g\ge 2$ and $z\in\Sigma$ a marked point. We prove that the subgroup of the mapping class group $\mathrm{Map}(\Sigma,z)$ corresponding to the fundamental group $\pi_1(\Sigma,z)$ of the closed surface does not lift to the group of diffeomorphisms of $\Sigma$ fixing $z$. As a corollary, we show that the Atiyah–Kodaira surface bundles admit no invariant flat connection, and obtain another proof of Morita’s non-lifting theorem.

Keywords: Mapping class group, lifting problem, flat connection, surface bundle

Bestvina Mladen, Church Thomas, Souto Juan: Some groups of mapping classes not realized by diffeomorphisms. Comment. Math. Helv. 88 (2013), 205-220. doi: 10.4171/CMH/283