# Some groups of mapping classes not realized by diffeomorphisms

### Mladen Bestvina

University of Utah, Salt Lake City, USA### Thomas Church

Stanford University, USA### Juan Souto

Université de Rennes 1, France

## Abstract

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.

## Cite this article

Mladen Bestvina, Thomas Church, Juan Souto, Some groups of mapping classes not realized by diffeomorphisms. Comment. Math. Helv. 88 (2013), no. 1, pp. 205–220

DOI 10.4171/CMH/283