For a surface S, Thurston asked if the natural surjection Homeo(S) → π_0 Homeo(S) splits, i.e. if there a natural action of the mapping class group Mod(S):= π_0 Homeo(S) on S. Markovic showed that no such action exists. On the other hand, there is a natural action of Mod(S) on the unit tangent bundle of S. More generally, for a 3-manifold M that fibers as a circle bundle over S, there is natural surjection Homeo(M) → Mod(S). We study when this surjection splits. This is joint work with Lei Chen and Alina al Beaini.