Mapping Class Groups of Circle Bundles over a Surface

Abstract

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