Let $S$ be a surface of genus $2$. What is a degree $2$ covering of $S$ by a surface $S'$ of genus $3$?
EDIT. I am looking for an explicit covering map.
2026-03-28 03:27:46.1774668466
What is a degree $2$ covering of $S$ by a surface $S'$ of genus $3$?
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Do you mean orientable surfaces? If so, then think of the genus 3 surface like this, with an axis going through the center of its middle hole:
The group $\mathbb Z/2\mathbb Z$ acts freely and properly discontinuously on this surface by rotating it 180° about the drawn axis. The quotient space under this action is an orientable surface of genus 2, and the quotient map is a covering map of degree $|\mathbb Z/2\mathbb Z| = 2$. Here I've used a general fact about group actions and covering maps which you can find on pg. 72 of Hatcher's book on algebraic topology, freely available on his website.
Note. Drawing a picture is not rigorous, but with this picture in mind, it is not hard to write down an explicit covering map.