  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
$S^1\rightarrow X\rightarrow S_g$. There is an exact sequence $1\rightarrow
H^1(S_g)\rightarrow Mod(X)\rightarrow Mod(S_g)\rightarrow1$. We relate this to
the Birman exact sequence and determine when this sequence splits.
