<p>In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.
For an orientable surface S the Euler characteristic χ(S) is
where g is the genus (the number of handles), 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</p>