Surface mapping class group actions on 3-manifolds

Abstract

For each circle bundle $S^1\to X\to {\Sigma }_g$ over a surface with genus $g\ge 2$, there is a natural surjection $\pi :{\mathrm{Homeo}}^{+}(X)\to \mathrm{Mod}({\Sigma }_g)$. 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 Chen and Tshishiku (2023). In this paper, we show that this homomorphism does not split in many cases where $\chi ({\Sigma }_g)$ divides $e(X)$.