= Double cover of the sphere with four branch values
{title2=$d=2,\quad R_f=4,\quad g_X=1$}
Suppose a nonconstant <holomorphic map> $f:X\to\mathbb P^1$ from a <connected> <compact Riemann surface> has <degree of a holomorphic map> two and exactly four distinct <branch values of a holomorphic map>. The <valency theorem> forces each of their <fibres> to contain one <ramification point> with $e_p=2$, a <simple ramification point of a holomorphic map>. Hence
$$
2g_X-2=2(-2)+4(2-1)=0,\qquad g_X=1.
$$
This determines the <genus> without recognizing a <torus> in a drawing.
A concrete realization is the <compact Riemann surface> obtained by adding the two points at infinity to the <even-degree hyperelliptic model> given by the <algebraic curve>
$$
w^2=P(z)=(z+2)(z+1)(z-1)(z-2).
$$
Projection to $z$ has <degree of a holomorphic map> two. At a <simple zero> $a$ of $P$, $w$ is a <local coordinate> and $z-a=w^2/P'(a)+O(w^4)$, so $e_p=2$. At infinity set $u=1/z$ and $s=w/z^2$: then $s^2=(1-4u^2)(1-u^2)$. There are two points with $u=0$, $s=\pm1$, each unramified since $u$ is a <local coordinate>. The four finite roots are therefore all the <branch values of a holomorphic map>.<Geometry of Surfaces by Nigel Hitchin (B3a course 2013)>[Section 3.3, printed pp. 41-43: local coordinates at simple roots, two points at infinity for even polynomial degree, and ramification count]{ref}
\Image[/double-sphere-cover-ramification-cell-count.png]
{title=Cell-count schematic for a double cover of the sphere with four simple branch values: four vertices, twelve edges and eight faces upstairs}
Choose a <triangulation of a surface> on the <Riemann sphere> with these four <branch values of a holomorphic map> as vertices, combinatorially the boundary of a <tetrahedron>. Its counts are $V=4,E=6,F=4$. Upstairs, each vertex has one preimage and each open edge and face has two lifts: $V_X=4,E_X=12,F_X=8$, giving <Euler characteristic> $4-12+8=0$. This is a cell schematic, not a geometric immersion of $X$. The <Triangulation proof of the Riemann-Hurwitz formula> applies the same counting idea after refining a general <triangulation of a surface> around the <branch values of a holomorphic map>.<Geometry of Surfaces by Nigel Hitchin (B3a course 2013)>[Section 3.2, Theorem 3.2 and its triangulation proof, printed pp. 38-39]{ref}
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