CounterexampleConstruction: AI-adaptedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Complex conjugation is a homeomorphism of the Riemann sphere that is not holomorphic

Statement refuted

Every self-homeomorphism of the Riemann sphere is holomorphic.

Facts & Assumptions

Given: Complex conjugation $κ (z) = \overline{z}$ on $C$ with $κ (\infty) = \infty$ .

[L1]

Stereographic projection identifies the Riemann sphere homeomorphically with the unit sphere (Stereographic projection identifies the Riemann sphere with the unit two-sphere).

Counterexample

technique · direct

1.1

The map $κ$ is continuous, involutive, and fixes $\infty$ , so it is a self-homeomorphism of $\hat{C}$ ; under [L1] it is the reflection of the unit sphere across the $x z$ -plane.

L1given

2.1

At $0$ , the complex difference quotient along real increments equals $1$ while along imaginary increments $i t$ it equals $- 1$ , so the complex derivative does not exist there. Thus this sphere homeomorphism is not holomorphic.

givenalgebra∎

Depends on

Used by

FALSE: every self-homeomorphism of the Riemann sphere preserves the cross-ratio False statement

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §§2.2-3.5 (standard reference, not scraped)
Matthias Weber, Complex Analysis, Ch. 1 §§1.3-1.4 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §§1-2 (standard reference, not scraped)