Surface mapping class group actions on 3-manifolds

For each circle bundle $S^1\to X\to\Sigma_g$ over a surface with genus $g\ge2$, there is a natural surjection $\pi:Homeo^+(X)\to Mod(\Sigma_g)$. When $X$ is the unit tangent bundle $U\Sigma_g$, it is well-known that $\pi$ splits. On the other hand $\pi$ does not split when the Euler number $e(X)$ is not divisible by the Euler characteristic $\chi(\Sigma_g)$ by work of the second two authors. In this paper we show that this homomorphism does not split in many cases where $\chi(\Sigma_g)$ divides $e(X)$.