I have a non-constant degree two map between Riemann surfaces $R$ and $S$, $f: R \to S$. I'm trying to find a holomorphic homeomorphism $\tau: R \to R$ such that $f(\tau) = f$ and $\tau^2$ is the identity.
I think I can define the correct map but I'm struggling to show it is a homeomorphism and holomorphic. Here is what I have so far:
- $f$ is degree 2 so from the degree formula this means that any point $s \in S$ can have at most 2 pre-images.
- I think we should say given $r \in R$, $\tau$ should send $r$ to it's pre-image friend $r'$ (where $f(r) = f(r')$) and if $r$ is a ramification point then $\tau$ should do nothing. This would certainly satisfy the properties required of $\tau$ and it's defined on the whole of $R$.
- To show this $\tau$ is holomorphic I understand we need to show it is holomorphic as a map in local coordinates. So we pick a neighbourhood $U$ small enough of $r$ so that $f$ can be written in local form in $U$. I was thinking of somehow using $f$ to help us define a local coordinate in which $\tau$ is holomorphic and homeomorphic but I'm struggling to get anywhere with this.
Thanks for any help