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:\operatorname{Homeo}^{+}(X)\to\operatorname{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) .