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Let $M$ and $\widetilde{M}$ be two Seifert fibered three manifolds. Suppose that there exists a covering projection $p: \widetilde{M} \to M$ preserving the Seifert structure. What is the relation between the Euler numbers $e \left( \widetilde{M} \right)$ and $e(M)$ of the two Seifert manifolds?

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  • $\begingroup$ This depends on two parameters: degree $d_1$ of the covering of the generic fiber and the degree $d_2$ of the covering between the bases (understood in the orbifold sense). Then $e(\tilde M)= e(M)d_2/d_1$, I think. $\endgroup$ Aug 25, 2015 at 21:04

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Let's say that a regular fiber of $M$ is covered by $u$ regular fibers of $\widetilde{M}$ and restriction of $p$ to each regular fiber has order $v$. Then $e(\widetilde{M}) = e(M) u / v$.

I don't have any reference for that fact from top of my head but it is visible just by thinking of the geometric role of numbers $(\alpha_i, \beta_i)$ for each singular fiber. I'm sure it's written down somewhere in Neumann-Raymond's book.

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