Riemann–Hurwitz formula

For an orientable surface S the Euler characteristic χ(S) is

where g is the genus (the number of holes), since the Betti numbers are 1, 2g, 1, 0, 0, ... . In the case of an (unramified) covering map of surfaces

that is surjective and of degree N, we should have the formula

That is because each simplex of S should be covered by exactly N in S′ — at least if we use a fine enough triangulation of S, as we are entitled to do since the Euler characteristic is a topological invariant. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (sheets coming together).
Now assume that S and S′ are Riemann surfaces, and that the map π is complex analytic. The map π is said to be ramified at a point P in S′ if there exist analytic coordinates near P and π(P) such that π takes the form π(z) = zⁿ, and n > 1. An equivalent way of thinking about this is that there exists a small neighborhood U of P such that π(P) has exactly one preimage in U, but the image of any other point in U has exactly n preimages in U. The number n is called the ramification index at P and also denoted by e_P. In calculating the Euler characteristic of S′ we notice the loss of e_P − 1 copies of P above π(P) (that is, in the inverse image of π(P)). Now let us choose triangulations of S and S′ with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then S′ will have the same number of d-dimensional faces for d different from zero, but fewer than expected vertices. Therefore we find a "corrected" formula

(almost all P have e_P = 1, so this is quite safe). This formula is known as the Riemann–Hurwitz formula and also as Hurwitz's theorem.