# Mapping Class Groups and torus (JSJ) decomposition of closed 3-manifolds

I am wondering if some intuitive relation exists between Mapping Class Group (MCG) of a 3-manifold (assume "simple" enough manifolds: closed,compact,irreducible, orientable, non-hyperbolic) and its JSJ decomposition? More generally, I am thinking along some generalization of Dehn-Nielsen-Baer theorem to 3-manifolds: in 2D there exists a minimal set of loops such that Dehn Twists along those loops generates the MCG. In 3D (again, "simple" enough manifolds) does there exist a minimal set of incompressible surfaces whose Dehn twisting generates the MCG? Thank you

The natural definition of a higher Dehn twist at a surface $F$ is to identify a neighborhood of $F$ homeomorphic to $F\times\left[0,1\right]$, take a loop $\phi_t$ in $Homeo(F)$ based at $id$, and consider the map $f\colon M\to M$ which is $\phi_t$ on $F\times\left\{t\right\}$ and the identity outside the neighborhood.
For a surface of genus $\ge 2$, the components of $Homeo(F)$ are contractible, so the Dehn twist at such a surface is homotopic to the identity. Similarly, Dehn twists at 2-spheres have order 2 because of $\pi_1SO(3)=\left\{\pm1\right\}$. Thus only Dehn twists at tori can be interesting.
For a closed Haken manifold, Johannson proved that the group generated by Dehn twists at tori has finite index in the mapping class group. (In particular the mapping class group of atoroidal Haken $3$-manifolds is finite.) For a survey you can look at Waldhausen's paper "Recent results on sufficiently large 3-manifolds", the result is the very last corollary on page 37.
• Well, $Homeo(F_{g\ge 2})$ is not trivial, just its components are contractible. – ThiKu Aug 14 '16 at 4:30