# 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

## 1 Answer

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.

There are some results for non-Haken manifolds, e.g., Gabai-Kazez have considered atoroidal manifolds with genuine laminations. But the general case seems to be still open(?).

• Thank you for the reference. I indeed did not know that Homeo(F) is trivial for genus >1 surfaces. Do you happen to know if, for the particular Haken manifold of a surface bundle over a circle (lets say surface with genus>1), full MCG (orientation preserving) be generated by Dehn twist of incompressible surfaces? What would these surfaces be in this particular case? – SKShukla Aug 14 '16 at 1:21
• Well, $Homeo(F_{g\ge 2})$ is not trivial, just its components are contractible. – ThiKu Aug 14 '16 at 4:30
• sorry, poor choice of words. Just meant they would generate the trivial element in MCG of the total 3-manifold. – SKShukla Aug 14 '16 at 4:47
• As said you have to look at Dehn twists at tori if you want something of infinite order. (It is actually known that the mapping class group of atoroidal irreducible 3-manifolds is always finite, whether the manifolds are Haken or not.) So in the case of surface bundles you have to look at ones with reducible monodromy. A incompressible torus arises as the mapping torus of an invariant curve. You can then compute the homology of the mapping torus and the action of the Dehn twist on the homology group. I guess in many cases it will have infinite order already for its action on the homology. – ThiKu Aug 14 '16 at 5:07