Riemann-Hurwitz Formula

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Introduction

For a compact, connected, orientable surface $S$, the Euler characteristic $\chi(S)$ is

$\chi(S)=2-2g$,

where g is the genus. This follows, as the Betti numbers are $1, 2g, 1, 0, 0, \dots$.

For the case of an (unramified) covering map of surfaces

$\pi\colon S' \to S$

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

$\chi(S') = N\cdot\chi(S).$

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):

Theorem

Now assume that $S$ and $S'$ are Riemann surfaces, and that the map $\pi$ is complex analytic.

The map $\pi$ 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 $π(z) = z^n$, 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 is 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

$\chi(S') = N\cdot\chi(S) - \sum_{P\in S'} (e_P -1) $

or as it is also commonly written, using that $\chi(X) = 2 - 2g(X)$ and multiplying through by $-1$:

$2g(S')-2 = N\cdot(2g(S)-2) +\sum_{P\in S'} (e_P -1) $

(all but finitely many $P$ have $e_P$ = 1, so this is quite safe).

Other form

$\chi(S')- r = N \cdot (\chi(S) - b) $

where $r$ is the number points in $S'$ at which the cover has nontrivial ramification (ramification points)

and $b$ is the number of points in $S$ that are images of such points (Branch points).

Indeed, to obtain this formula, remove disjoint disc neighborhoods of the branch points from $S$ and disjoint disc neighborhoods of the ramification points in $S' $ so that the restriction of $ \pi $ is a covering. Then apply the general degree formula to the restriction, use the fact that the Euler characteristic of the disc equals 1,

and use the additivity of the Euler characteristic under connected sums.

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Also known as

The Riemann-Hurwitz Formula is also known as Hurwitz's Theorem.

Source of Name

This entry was named for Georg Friedrich Bernhard Riemann and Adolf Hurwitz.

Sources

• 1977: Robin Hartshorne: Algebraic Geometry: $\S \text {IV}.2$
• 2006: Jurgen Jost: Compact Riemann Surfaces