Mapping class groups of circle bundles over a surface

In this paper, we study the algebraic structure of mapping class group $Mod(X)$ of 3-manifolds $X$ that fiber as a circle bundle over a surface $S^1\to X\to S_g$. There is an exact sequence $1\to H^1(S_g)\to Mod(X)\to Mod(S_g)\to 1$. We relate this to the Birman exact sequence and determine when this sequence splits.