= Genus of a finite cover of a closed orientable surface
{title2=$g'=1+n(g-1)$}
A connected $n$-sheeted <covering space> of a <closed orientable surface> of genus $g$ has genus $1+n(g-1)$. Lifting a finite cell decomposition gives $n$ cells above every base cell, so its <Euler characteristic> is $n(2-2g)$. Equating this with $2-2g'$ gives the formula. For $g\ge1$, every positive $n$ occurs: send one handle generator of the <fundamental group> to $1$ in $\mathbb Z/n$ and all remaining generators to zero, then use the <classification of connected covering spaces> for the kernel of that surjection.
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