Mapping class groups of circle bundles over a surface

TL;DR

This paper investigates the algebraic structure of the mapping class group of 3-manifolds that are circle bundles over surfaces, analyzing exact sequences and conditions for splitting.

Contribution

It provides a detailed analysis of the mapping class group of circle bundle 3-manifolds and determines when the associated exact sequence splits.

Findings

Identifies the exact sequence relating $Mod(X)$ and $Mod(S_g)$

Relates the sequence to the Birman exact sequence

Determines conditions under which the sequence splits

Abstract

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.