# Mapping class groups of small Seifert-fibred 3-manifolds

Are computations of the mapping class groups of small Seifert-fibred 3-manifolds recorded in some convenient location?

For most Seifert manifolds working out the mapping class group is easy-enough (at least, reducing it to a 2-dimensional mapping class group is easy) since diffeomorphisms usually are isotopic to fibre-preserving diffeomorphisms. But for the small Seifert-fibred manifolds that isn't always the case.

For lens spaces and general spherical 3-manifolds this was worked out by Darryl McCullough and others, about 10 years ago.

• McCullough, Darryl. Isometries of elliptic 3-manifolds. J. London Math. Soc. (2) 65 (2002), no. 1, 167–182.

For the purpose of this question, "mapping class group" of a 3-manifold means the diffeomorphism group modulo the subgroup of diffeomorphisms isotopic to the identity, i.e. $\pi_0 Diff(M)$.

If this is in one of the standard references like Orlik's book, please let me know. I looked over it briefly but it looked to me like Orlik's book does not cover this, at least not explicitly. I imagine one could derive the computation from any suitably-detailed proof of the uniqueness of Seifert-fiberings, but I haven't worked out the details myself.

• See section 9 of this paper for discussion and references in the case of a base orbifold hyperbolic. projecteuclid.org/euclid.jdg/1361800869 I think all of the cases of Nil and Euclidean geometries are Haken, so should be covered by theorems of Hatcher. Nov 16, 2013 at 23:09