Closed orientable surface

A closed orientable surface of genus $g$ is homeomorphic to a sphere with $g$ handles. Its Euler characteristic is $2-2g$.

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 $2-2g^{\prime }$ gives the formula. For $g\ge 1$, 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.