# Mapping class groups of Haken Seifert 3-manifolds (not small)

This is a somewhat broad question. I would like to get an idea of what is known about mapping class groups (MCG) of a Seifert manifold $$M$$, in particular how to systematically compute it.

I want to restrict to Haken ones for now, and it is known that $$\text{MCG}(M)$$ is isomorphic to $$\text{Out}(\pi_1(M))$$ (c.f. page 33 of https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/3-manifold-groups-final-version-031115).

My main focus is $$M$$ over an $$S^2$$ base, namely $$S^2\left(b;\frac{a_1}{b_1},\dots,\frac{a_n}{b_n}\right)$$ (sorry for this non-standard notation). A potentially useful reference is https://arxiv.org/abs/math/0010077 by D. McCullough, whose Table 4 contains only $$S^2(2,2,m), S^2(2,3,3), S^2(2,3,4), S^2(2,2,5)$$. Their MCGs are known according to Section 3 in the same paper.

I would like to know at least the MCGs of Seifert manifolds $$S^2\left(\frac{a_1}{b_1},\frac{a_2}{b_2},\frac{a_3}{b_3}\right)$$, which are beyond the scope of the paper by McCullough.

The last case you are interested in is essentially covered by a paper of McCullough and Soma. For such manifolds (other than three-sphere geometry) the inclusion of the isometry group into the diffeomorphism group is a homotopy equivalence. Thus the mapping class group is finite, and can be deduced from the symmetries of the base, together with a possible reflection of the fiber. (The assumption of Haken or non-Haken does not play a role.)

When there are more cone points the problem appears to be more delicate. Hopefully an expert will weight in; here is my understanding of the situation.

Suppose that $$M$$ is a Seifert fibered space with a unique Seifert fibering.

1. Thus all mapping classes have a representative that preserves the fibering. (There is a subtle point here. I think we want the group of all homeomorphisms of $$M$$ to deformation retract to the subgroup of fibre-preserving homeomorphisms.)
2. Drill out a small open fibered neighbourhood $$U$$ of a generic fibre to obtain $$M' = M - U$$.
3. Thus $$M'$$ is a surface bundle with fiber $$F'$$ over the circle. This gives us a branched covering from $$F'$$ to the (once-punctured) orbifold base $$B'$$. Compute the monodromy of this covering.
4. Compute the mapping class group of the base $$B'$$.
5. Compute the finite index subgroup that lifts to give mapping classes of $$F'$$.
6. Each lifted mapping class gives a homeomorphism of $$M'$$; these act on $$\partial M'$$. There is a (index two) subgroup that preserves the meridional slope of $$U$$.

This feels like an approachable, but definitely non-trivial, problem.