Mapping class groups of circle bundles over a surface

In this talk, we study the algebraic structure of mapping class group $Mod(M)$ of 3-manifolds $M$ that fiber as a circle bundle over a surface. We prove an exact sequence, relate this to the Birman exact sequence, and determine when this sequence splits. We will also discuss the Nielsen realization problem for such manifolds and give a partial answer. This is joint work with Bena Tshishiku and Alina Beaini.