Proof of the Riemann-Hurwitz Formula

The Riemann-Hurwitz formula relates the Euler characteristics (and hence the genera) of Riemann surfaces connected by a branched covering.

Statement

Theorem10.10Riemann-Hurwitz formula (restated)

Let $f: S \to T$ be a holomorphic map of degree $n$ between compact Riemann surfaces. Then

$2g_S - 2 = n(2g_T - 2) + R$

where $R = \sum_{p \in S}(e_p - 1)$ is the total ramification and $e_p$ is the ramification index at $p$ .

Proof

Step 1: Triangulate the target.

Choose a triangulation of $T$ such that every branch value of $f$ is a vertex. Let the triangulation have $V$ vertices, $E$ edges, and $F$ faces. Then $\chi(T) = V - E + F = 2 - 2g_T$ .

Step 2: Pull back the triangulation.

The map $f$ lifts the triangulation to $S$ . Away from branch points, $f$ is a local homeomorphism, so:

Each face of $T$ has exactly $n$ preimages $\Rightarrow$ $F_S = nF$ .
Each edge of $T$ has exactly $n$ preimages $\Rightarrow$ $E_S = nE$ .
For vertices: a vertex $q \in T$ has preimages $p_1, \ldots, p_k$ with $\sum_{j=1}^k e_{p_j} = n$ (the total local degree is $n$ ). So the total number of preimage vertices is $V_S = \sum_{q \in T}\sum_{p \in f^{-1}(q)} 1$ .

For each vertex $q$ of $T$ : $|f^{-1}(q)| = n - \sum_{p \in f^{-1}(q)}(e_p - 1)$ .

Summing over all vertices: $V_S = nV - R$ where $R = \sum_{p \in S}(e_p - 1)$ .

Step 3: Compute the Euler characteristic.

$\chi(S) = V_S - E_S + F_S = (nV - R) - nE + nF = n(V - E + F) - R = n\chi(T) - R.$

Step 4: Convert to genus.

Since $\chi = 2 - 2g$ :

$2 - 2g_S = n(2 - 2g_T) - R.$

Rearranging: $2g_S - 2 = n(2g_T - 2) + R$ . $\blacksquare$

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Verification and Examples

ExampleVerification for $z^n$

$f(z) = z^n: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ has degree $n$ , $g_S = g_T = 0$ . Branch points: $z = 0$ ( $e = n$ ) and $z = \infty$ ( $e = n$ ). Total ramification: $R = (n-1) + (n-1) = 2n - 2$ .

Check: $2(0) - 2 = n(2(0)-2) + 2n - 2$ , i.e., $-2 = -2n + 2n - 2 = -2$ .

ExampleDouble cover of the torus

Let $f: S \to T$ be a double cover ( $n = 2$ ) of a torus ( $g_T = 1$ ) with $4$ simple branch points ( $e = 2$ at each). Then:

$2g_S - 2 = 2(2-2) + 4 = 4 \implies g_S = 3.$

So a double cover of a torus branched at $4$ points has genus $3$ .

ExampleGenus bound for coverings

For an unramified covering ( $R = 0$ ) of degree $n$ : $g_S - 1 = n(g_T - 1)$ . Since $g_S \geq 0$ : $n \leq 1/(1 - g_T)$ if $g_T = 0$ (only $n = 1$ ); any $n$ if $g_T \geq 1$ .

For $g_T = 1$ : $g_S = 1$ for unramified covers of any degree (unramified covers of tori are again tori).

Generalization

RemarkOrbifold Euler characteristic

The Riemann-Hurwitz formula can be expressed using the orbifold Euler characteristic: $\chi^{\text{orb}}(T) = \chi(T) - \sum_q (1 - 1/e_q)$ where the sum is over branch values with local ramification $e_q$ . Then $\chi(S) = n \cdot \chi^{\text{orb}}(T)$ . This viewpoint is fundamental in the theory of orbifolds and uniformization of surfaces with cone points.