Surface mapping class group actions on 3-manifolds

Abstract

For each circle bundle $S^1\to X\to \hat{I}{\pounds }_g$ over a surface with genus $g\ge 2$, there is a natural surjection $\ddot{I}€:Home{o}^{+}(X)\to Mod(\hat{I}{\pounds }_g)$. When $X$ is the unit tangent bundle $U\hat{I}{\pounds }_g$, it is well-known that $\ddot{I}€$ splits. On the other hand $\ddot{I}€$ does not split when the Euler number $e(X)$ is not divisible by the Euler characteristic $\ddot{I}‡(\hat{I}{\pounds }_g)$ by work of the second two authors. In this paper we show that this homomorphism does not split in many cases where $\ddot{I}‡(\hat{I}{\pounds }_g)$ divides $e(X)$.

15 pages