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Let $M_g$ and $M_h$ be closed orientable 3-manifolds of genus $g$ and $h$ respectively and suppose that $M_g$ is an $n$-sheeted cover of $M_h$. Is there a formula that would allow us to compute $g$ if we knew the values of $h$ and $n$?

I know there is a formula for closed orientable surfaces and I was wondering if there was a result for $3$-manifolds.

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Heegaard genus is not very constructive and is very difficult to control. Obviously, there cannot be a formula in terms of just $n$ and $h$. It suffices to consider the double coverings $S^3\to\Bbb R\mathrm{p}^3$ and $S^1\times S^2\to S^1\times S^2$.

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  • $\begingroup$ There is no formula, but there might be a bound. $\endgroup$
    – Igor Rivin
    Jan 28, 2015 at 23:51

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