Let $S$ be a closed orientable surface. For any positive integer $n$, is there a connected $n$-sheeted covering space of $S$? This is certainly not true if $S=S^2$ because $S^2$ is simply-connected. The result is true for $S=T^2$, since $\pi_1(T^2)=\Bbb Z^2$ has an index $n$ subgroup for each $n$. However I cannot handle the other cases. I know that this is equivalent to finding an index $n$ subgroup of $\pi_1(S)$ but the $\pi_1$ for the other surfaces are quite complicated, I think. Any hints?
Is there a connected $n$-sheeted covering space of a closed orientable surface for any positive integer $n$
algebraic-topologycovering-spacesfundamental-groupssurfaces
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Best Answer
Yes. See the picture for an example that a surface of genus 3 got a 5-sheeted covering:
This lay out works for any surfaces of genus $\geq 1$ and any $n$.