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.



Genus of Covering Space of 3-Manifold
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 $S^1\times S^2\to S^1\times S^2$.


