Some groups of mapping classes not realized by diffeomorphisms

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.