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Riemann–Hurwitz for the sphere power map

Example

Assume the Axiom of Choice. For every $n \geq 1$ the power map $f : \hat{C} \to \hat{C}, f (z) = z^{n} (z \in C), f (\infty) = \infty,$ is a holomorphic map of the Riemann sphere of degree $n$ , with $e_{0} (f) = n$ , $e_{\infty} (f) = n$ and $e_{z} (f) = 1$ for every $z \in C^{\times}$ . Riemann–Hurwitz for $f$ therefore reads $- 2 = - 2 n + 2 (n - 1),$ both sides equal to $- 2$ , with the genus of the sphere equal to $0$ . For $n = 1$ the map is the identity, all indices equal $1$ , and there is no ramification.

Facts & Assumptions

Given: An integer $n \geq 1$ and the map $f (z) = z^{n}$ on $C$ , $f (\infty) = \infty$ .

[F1]

The standard charts of $\hat{C}$ are $ϕ_{0} (z) = z$ on $U_{0} = \hat{C} ∖ {\infty}$ and $ϕ_{\infty} (z) = 1 / z$ on $U_{\infty} = \hat{C} ∖ {0}$ , with transition $w \mapsto 1 / w$ on the overlap (The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity); these charts make $\hat{C}$ a Riemann surface (Atlases on the sphere, plane, disc and annulus) that is compact Hausdorff, being the one-point compactification of $C$ (The Riemann sphere is the published one-point compactification of the complex plane).

[F2]

A map of Riemann surfaces is holomorphic when its chart expressions are holomorphic; in the standard charts a map fixing $\infty$ is holomorphic at infinity exactly when the expression $u \mapsto 1 / f (1 / u)$ is holomorphic at $0$ (Holomorphic maps and meromorphic functions on Riemann surfaces, The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity).

[F3]

A complex polynomial $P (z) = \sum_{k} a_{k} z^{k}$ is entire with $P^{'} (z) = \sum_{k} k a_{k} z^{k - 1}$ (Complex polynomials are entire with the power-rule derivative, and rational functions are holomorphic wherever their denominator is nonzero); in particular $z \mapsto z^{n}$ is entire with derivative $n z^{n - 1}$ , which vanishes only at $z = 0$ when $n \geq 2$ and nowhere when $n = 1$ .

[F4]

Ramification index: $e_{x} (f)$ is the unique $e \geq 1$ with the chart expression $z \mapsto z^{e}$ in suitable centred charts, and $e_{x} (f) = 1$ exactly when $f$ is a local biholomorphism at $x$ ; $x$ is a critical point when $e_{x} (f) > 1$ and $f (x)$ is then a branch value (Ramification index, ramification order and branch value, Biholomorphic maps between complex domains).

[F5]

Degree of a proper nonconstant holomorphic map: it is onto with finite fibres and has a degree $d = \sum_{x \in f^{- 1} (y)} e_{x} (f)$ independent of $y$ (Degree of a proper holomorphic map of Riemann surfaces).

[F6]

For $w \neq 0$ the equation $z^{n} = w$ has exactly $n$ distinct solutions, the $n$ -th roots of $w$ ; the $n$ -th roots of unity are exactly $\exp (2 π i k / n)$ , $0 \leq k < n$ (The $n$ -th roots of a complex number and the $n$ distinct roots of unity for every $n \geq 1$ ).

[F7]

Stereographic projection identifies the Riemann sphere $\hat{C}$ homeomorphically with $S^{2}$ (Stereographic projection identifies the Riemann sphere with the unit two-sphere); by the genus definition, $g (\hat{C}) = 0$ and $χ (\hat{C}) = 2$ (Genus and Euler characteristic of a compact Riemann surface).

[F8]

Riemann–Hurwitz: for a nonconstant holomorphic map $f : X \to Y$ of compact connected Riemann surfaces of degree $d$ , $2 g (X) - 2 = d (2 g (Y) - 2) + \sum_{x \in X} (e_{x} (f) - 1)$ (Riemann–Hurwitz formula for compact Riemann surfaces).

[F9]

The Axiom of Choice (The Axiom of Choice).

Verification

technique · direct

1.1F2F3

( $f$ is a holomorphic nonconstant self-map of the sphere.) In the chart $ϕ_{0}$ the expression of $f$ is $z \mapsto z^{n}$ , entire by [F3]; at infinity, using the source chart $u = 1 / z$ and the target chart $v = 1 / f (z)$ , the expression is $v = u^{n}$ for $u \neq 0$ , which extends holomorphically to $u = 0$ with value $0 = ϕ_{\infty} (f (\infty))$ ; hence $f$ is holomorphic on $\hat{C}$ by [F2]. It is nonconstant: for $n = 1$ it is the identity and for $n \geq 2$ the values $0$ and $1$ differ.

1.2F1F5

( $f$ is proper.) $\hat{C}$ is compact and $f$ is continuous [F1]; for every compact $K \subseteq \hat{C}$ the preimage $f^{- 1} (K)$ is closed in the compact space $\hat{C}$ , hence compact. Thus [F5] applies to $f$ .

1.3F3F4

(The ramification indices.) At $0$ the centred charts are the standard charts near $0$ , and the chart expression is $z \mapsto z^{n}$ , so $e_{0} (f) = n$ by [F4]. At $\infty$ the centred charts $(u, 1 / u)$ on the source and $(v, 1 / w)$ on the target give the expression $u \mapsto u^{n}$ , so $e_{\infty} (f) = n$ ; for $n = 1$ both statements read $e = 1$ and there is no critical point. For $a \in C^{\times}$ the chart expression near $a$ is $z \mapsto z^{n}$ with derivative $n z^{n - 1}$ nonzero at $a$ by [F3], so $f$ is a local biholomorphism at $a$ and $e_{a} (f) = 1$ by [F4]; such $a$ is therefore not a critical point.

2.1F5F6step 1.3

(The degree is $n$ .) By [F5] and step 1.2 the degree equals $\sum_{x \in f^{- 1} (1)} e_{x} (f)$ . The solutions of $z^{n} = 1$ are the $n$ distinct $n$ -th roots of unity, all in $C^{\times}$ , by [F6]; each has index $1$ by step 1.3, so $d = n$ .

3.1F7F8step 1.3step 2.1

(Riemann–Hurwitz reads $- 2 = - 2 n + 2 (n - 1)$ .) Apply [F8] to $f$ , which is nonconstant holomorphic of degree $d = n$ between compact connected Riemann surfaces by steps 1.1, 1.2 and 2.1: $2 g (\hat{C}) - 2 = n (2 g (\hat{C}) - 2) + \sum_{x} (e_{x} (f) - 1)$ . By the stereographic homeomorphism and genus definition in [F7], $g (\hat{C}) = 0$ . For $n \geq 2$ , step 1.3 gives exactly two ramification points, $0$ and $\infty$ , each with $e = n$ , so the sum is $2 (n - 1)$ ; for $n = 1$ there is no ramification and the empty sum is also $2 (n - 1) = 0$ . Substituting gives $- 2 = - 2 n + 2 (n - 1)$ , an identity of integers.

4.1F7F8F9step 3.1∎

(Conclusion and choice.) All indices, the degree and the ramification sum are computed from the explicit charts, so this example is choice-free; the Axiom of Choice is inherited only through the genus interface [F7] used in [F8], as [F9] records.

Remarks

The power map is the simplest nontrivial Riemann–Hurwitz identity: for $n \geq 2$ the two critical points $0$ and $\infty$ each contribute $n - 1$ , so the total deficit is $2 (n - 1)$ , and in Euler-characteristic form the count gives $χ = d χ (\hat{C}) - \sum (e_{x} - 1) = 2 n - 2 (n - 1) = 2$ , the Euler characteristic of a sphere again — as it must be, since the source is the sphere. Criticality is independent of the coordinate choices: at both $0$ and $\infty$ , centred source and target coordinates give the same local model $u \mapsto u^{n}$ , with ramification index $n$ . For the same computation in the general $P^{1}$ setting with the local model $z \mapsto z^{n}$ see morphism projective line power map; the present example stays in the sphere charts used throughout this pair.

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Sources

Eduard Looijenga, Riemann Surfaces (2007) (standard reference, not scraped)
Curtis T. McMullen, Riemann Surfaces, Math 213b course notes (2026) (standard reference, not scraped)
Vladimir Hinich, Riemann Surfaces, lecture 7, §8.5 (standard reference, not scraped)