# Seifert fiberable manifolds with several Seifert fiberings

I have a question on Theorem 2.3 on page 34 of Hatcher's notes on 3-manifolds: Hatcher: Notes on Basic 3-Manifold Topology.

Regarding the class d), it follows from Proposition 2.1 on page 31, that $M(0,0;1/2,-1/2,\alpha/\beta)$ is fiber-diffeomorphic to $M(0,0;1/2,-1/2,\alpha'/\beta')$ if and only if $\alpha/\beta=\pm \alpha'/\beta'$ if and only if $M(-1,0;\beta/\alpha)$ is fiber-diffeomorphic to $M(-1,0;\beta'/\alpha')$. But it is not clear to me, that for different $q,q'\in\mathbb{Q}$ with $q,q'\geq 0$ the manifolds $M(-1,0;q)$ and $M(-1,0;q')$ are not diffeomorphic and also not diffeomorphic to the manifolds under a),b),c),e), i.e. not diffeomorphic to the solid torus, the twisted $I$-bundle or the twisted $S_1$-bundle over the Klein bottle, or any lens space.

Is this true anyway? And do those manifolds under d) have a name (i.e. are known under a certain name like lens spaces)?

• did you tried to see their fundamental groups? Commented Feb 16, 2014 at 20:36

As explained on page 37 of the notes, a complete proof of the full classification of orientable Seifert manifolds (Theorem 2.2) is not given in the notes. What is missing is the classification of the manifolds that fiber over $S^2$ with exactly three multiple fibers. The statement here is that these Seifert manifolds are all distinguished by their fundamental groups (which are not cyclic so they are not lens spaces, $S^3$, or $S^1\times S^2$), and their fiberings are unique apart from the exceptions listed in part (d). A proof of this can be found in the reference given, namely Orlik's Springer Lecture Notes volume #291.
• @Werner Thumann: That is correct, each prism manifold has exactly two Seifert fiberings. This is an interesting contrast to lens spaces, $S^3$, and $S^1\times S^2$, each of which has infinitely many different Seifert fiberings. Commented Feb 19, 2014 at 15:15