# Mapping class group of certain 3-manifolds

Let $\xi : M^3 \to F$ be an orientable circle bundle over a closed orientable surface $F$ of genus $g \geq 2$. I am mostly interested to the case where the bundle $\xi$ is non-trivial. My question is about the mapping class group $\mathrm{MCG} (M) = \pi_0(\mathrm{Diff}_+(M))$. In particular, is this group generated by liftings of Dehn twists and bundle automorphisms?

Since you write ${\rm Diff}_+(M)$ you are probably assuming $M$ is orientable and diffeomorphisms of $M$ are orientation-preserving. Every diffeomorphism of $M$ can be isotoped to take fibers to fibers. This is proved using the assumption that the base surface $F$ has genus at least 2, so $M$ contains vertical incompressible tori (unions of fibers) and every incompressible torus can be isotoped to be vertical. If a diffeomorphism that takes fibers to fibers preserves orientations of the fibers then it is isotopic to a product of twists along vertical tori. These are generated by lifts of Dehn twists in $F$ and twists taking each fiber to itself, i.e., bundle automorphisms.
The other possibility is a diffeomorphism taking fibers to fibers but reversing their orientations, hence also reversing orientation of $F$ since the orientation of $M$ is preserved. Such diffeomorphisms certainly exist for the product bundle $M=F\times S^1$, and they also exist for any nontrivial bundle whose Euler class is even. Namely, start with an example for $F\times S^1$ and modify this by removing two vertical solid tori $V_1$ and $V_2$ that are interchanged by the diffeomorphism, then glue $V_1$ and $V_2$ back in by diffeomorphisms of their boundary tori preserving fibers and taking a slope $0$ curve to a slope $n$ curve, using the same $n$ in both cases. This gives a circle bundle with Euler class $2n$. The diffeomorphism of $M-(V_1\cup V_2)$ reversing orientations in fiber and base extends over the reglued solid tori since slopes in a torus $S^1\times S^1$ are preserved by a $180$ degree rotation that reflects each $S^1$ factor.
Added a few minutes later: It looks like a similar construction can be made more simply using one vertical solid torus instead of two, where this solid torus projects to a disk neighborhood of a fixed point of an orientation-reversing diffeomorphism of $F$. In this case there is no need to assume the Euler class is even.
• I understand, thank you! Yes, orientability of $M$ and orientation-preserving homeomorphisms are what I was thinking about. – Daniele Zuddas Jun 8 '17 at 6:54
• The theorem that diffeomorphisms of $M$ can be isotoped to take fibers to fibers is a special case of a theorem of Waldhausen in a two-part paper on graph manifolds in Inventiones 3-4 (1967-68). He worked in the PL category but the proof applies also in the smooth category if one uses Cerf's theorem that $\pi_0{\rm Diff}_+(S^3)=0$. Another classical exposition is in Jaco's Lectures on Three-Manifold Topology (1980), Theorem VI.18, which gives the analogous result for Seifert manifolds with non-empty boundary, leaving the closed case as an exercise (using similar methods). – Allen Hatcher Jun 9 '17 at 0:36