I'm trying to make a non-constant holomorphic map, $f$, between a 3 holed torus and a 2 holed torus, with no branch points.
Now I can see that $deg(f) = 2$ from the Riemann Hurwitz formula. So intuitively, I want to treat the three holed torus as two tori attached with handles and then have a 2-1 map from this surface into a 2 holed torus.
I'm having trouble actually writing down this map. Is there a nice way to embed a three holed torus? Analogously to how we can write a torus as $T = S^1 x S^1$ as a subset of $\mathbb{C}^2$
Thanks
Picture the standard depiction of $\Sigma_3$, where the three holes are in a row. Note that this surface has 180-degree rotational symmetry, i.e. $\mathbb Z/2\mathbb Z$ acts on it. Also note that this action is free (i.e. neither element of $\mathbb Z/2\mathbb Z$ fixes any point of $\Sigma_3$) and each orbit is discrete. As such, we can consider the quotient of $\Sigma_3$ by this action. This quotient turns out to be $\Sigma_2$, which we find by picturing the "right half" of $\Sigma_3$. There is a hole (the rightmost hole), and the right half is bound by two loops, which, when we pass to the quotient, are glued together, making another hole.
This construction generalizes to give a holomorphism $\Sigma_{mn+1} \to \Sigma_{m+1}$ for any nonnegative integers $m, n$.