# 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.

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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. –  Ian Agol Nov 16 '13 at 23:09

The determination of mapping class groups of small Seifert manifolds was completed by M. Boileau and J.-P. Otal in a paper in Invent. Math. 106 (1991), 85-107. They give references for cases previously done:

1. Lens spaces by Bonahon and Hodgson-Rubinstein.
2. Multiple fibers of orders (2,2,n) by Asano and Rubinstein.
3. Multiple fibers of orders (2,3,4) by Birman-Rubinstein.
4. Multiple fibers of orders (p,q,r) in most of the cases with infinite fundamental group by Scott.

This left only the cases (2,3,p) other than (2,3,4), and the cases (3,3,p). These are the cases treated in the Boileau-Otal paper. In particular this includes the Poincaré homology sphere, where the mapping class group happens to be trivial.

All this was done in the 1980s. (Boileau and Otal announced their results in a C.R. note in 1986.)

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Thanks Allen. I think I must have known this when I was a grad student, since I've scanned all those papers at some point. –  Ryan Budney Nov 17 '13 at 15:37