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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.

1 Answers1

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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:

enter image description here

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.

  • Hi @user33698, thanks for the answer! Is it possible you could elaborate on what you meant by "Of course, you can make this rigorous by writing down explicit maps."? –  Oct 23 '16 at 23:05
  • I just meant that drawing a picture and describing a group action on the picture is non-rigorous, but with this picture in mind, it isn't hard to explicitly describe the covering map once you've chosen how to represent these surfaces (say, by gluing the sides of certain polygons). –  Oct 23 '16 at 23:17