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

Question

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.

Answer 1

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.)

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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.