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I seek a reference for what is surely a well known basic result about Seifert fibered 3-manifolds. Namely they are all obtained by Dehn-surgery along a regular Seifert fiber (and the surgery slope is not that of the fiber) of a surface bundle or semi-bundle with periodic monodromy. In particular the Euler number is zero if and only if the manifold is orientable and the surgery is non-trivial . This is in the general case that includes fibered solid Klein bottles. Formally:

Suppose $M$ is a closed 3-manifold equipped with a Seifert fibration. Let $V$ be a solid torus in $M$ that is a union of circle fibers. Let $N$ be the submanifold of $M$ obtained by removing the interior of $V$. Then $N$ is a compact Seifert fiber space with boundary a torus.

There is a compact, connected, horizontal surface $F$ that is $2$-sided embedded in $N$, with $\partial F\subset\partial N$. Moroever, if $F$ is any such surface, and $\alpha$ a boundary component of $F$ then

  1. $N$ is a surface bundle, or semi-bundle, with (regular) fiber $F$.
  2. There is a finite sheeted covering space $p:F\times S^1\rightarrow N$ that is a bundle map.
  3. $p$ restricts to a homeomorphism on each component of $p^{-1}(F)$.
  4. The foliation of $N$ by circles is covered by this product foliation.
  5. $M$ is finitely covered by $\Sigma\times S^1$ iff $M$ is non-orientable, or $\alpha$ bounds a disc in $V$.
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Minus the solid Klein bottles, most of your argument is contained in Proposition 2.2 of Hatcher's notes on three-manifolds. Here is a link:

https://pi.math.cornell.edu/~hatcher/3M/3Mdownloads.html

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    $\begingroup$ Thanks. A minor modification of Hatcher's proof works in the general case. Still, it would be nice to have a reference for the general case. $\endgroup$ May 29 at 5:14

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