Concept

Riemann–Hurwitz formula

Summary

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 a compact, connected, orientable surface , the Euler characteristic is where g is the genus (the number of handles), since the Betti numbers are . In the case of an (unramified) covering map of surfaces that is surjective and of degree , we have the formula That is because each simplex of should be covered by exactly in , at least if we use a fine enough triangulation of , 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 and are Riemann surfaces, and that the map is complex analytic. The map 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 eP. In calculating the Euler characteristic of S′ we notice the loss of eP − 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 or as it is also commonly written, using that and multiplying through by -1: (all but finitely many P have eP = 1, so this is quite safe).

Official source

https://en.wikipedia.org/wiki/Riemann–Hurwitz_formula

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