Triangulation proof of the Riemann-Hurwitz formula
= Triangulation proof of the Riemann-Hurwitz formula
{c}
Triangulate the target with all branch values among its vertices. For a degree-$d$ map, every edge and face has $d$ lifts. Above a target vertex $v$, the number of lifted vertices is
$$
d-\sum_{p\mapsto v}(e_p-1),
$$
because $\sum_{p\mapsto v}e_p=d$. Thus
$$
\chi(X)=d\chi(Y)-\sum_p(e_p-1),
$$
which is equivalent to the <Riemann-Hurwitz formula>.