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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)?

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  • $\begingroup$ did you tried to see their fundamental groups? $\endgroup$
    – janmarqz
    Commented Feb 16, 2014 at 20:36

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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.

The manifolds of type (d) in Theorem 2.2 are among these manifolds. They are closed manifolds so they are not of type (a) or (b). They are also not of type (e) since they do not contain incompressible tori, as shown earlier in the notes. They are not of type (c) since their fundamental groups are noncyclic as noted above. Manifolds of type (d) have 2-sheeted covering spaces which are lens spaces, so they have finite fundamental group. Sometimes they are called prism manifolds, from a way of constructing them by identifying faces of a prism.

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  • $\begingroup$ Dear Prof. Hatcher. Thank you for your reply. Did I understand it correctly that each of those exceptional prism manifolds have exactly two Seifert fibrations? $\endgroup$
    – Werner Thumann
    Commented Feb 18, 2014 at 20:36
  • $\begingroup$ @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. $\endgroup$
    – Allen Hatcher
    Commented Feb 19, 2014 at 15:15

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