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It is well-known that Seifert fibered $3$--manifolds are geometric: they admit one of the Thurston geometries $S^2 \times R$, $R^3$, $H^2 \times R$, $S^3$, $Nil$, and $PSL(2,R)$. Furthermore, the converse is also true: every $3$--manifold admitting one of these 6 geometries has a Seifert fibration over a $2$--orbifold.

In the realm of $3$--orbifolds, it is still true that every Seifert fibered $3$--orbifold is geometric, with the same 6 geometries. However, the converse is false: there exist geometric $3$--orbifolds with one of the 6 "Seifert" geometries, which do not Seifert fiber. My question is:

What are the orientable $3$--orbifolds that have a Seifert geometry, but are not Seifert fibered? Do we know a complete list?

One example of this weird phenomenon is the figure-8 knot, labeled 3. This orbifold is Euclidean. But it is not Seifert fibered, because any order-3 singular locus must be vertical in a Seifert fibration. Thus, if there was a Seifert fibering of the orbifold, drilling out the singular locus would produce a Seifert fibration on the knot complement, which is absurd because the knot complement is hyperbolic.

More generally, Dunbar has classified the spherical $3$--orbifolds that are not Seifert fibered. There are 21 such examples in total:

What is known about the other 5 Seifert geometries?

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up vote 8 down vote accepted

From Three-dimensional orbifolds and their geometric structures, by Boileau, Maillot and Porti (p. 15):

Remark. There are orbifolds with geometry $E^3$ and $S^3$ which are not orbifold-Seifert fibered [52, 54].

[54] is this article of Dunbar's you quoted. [52] is Geometric orbifolds, of the same author.

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Thanks! But the link to "Geometric orbifolds" seems to be broken. If this is Dunbar's paper in Rev. Mat. Univ. Complut. Madrid, I would love to get access to a copy. – Dave Futer Jun 30 '11 at 18:47
I edited it just after I posted that message. Does it work now? – Maxime Bourrigan Jun 30 '11 at 18:48
Yes, the link works perfectly -- and it appears that the paper has exactly the answer I'm looking for. Much thanks! – Dave Futer Jun 30 '11 at 18:55

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