Is any homeomorphism from Riemann sphere to Riemann sphere Mobius transformation?

Question

Let $\hat{\mathbb{C}}$ be the Riemann sphere.

Let $f:\hat{\mathbb{C}}\rightarrow \hat{\mathbb{C}}$ be a homeomorphism.

Then, is $f$ a Mobius transformation?

Answer 1

Any conformal homeomorphism is a Moebius transformation; but there are many homeomorphisms of $S^2$ which are not conformal. Here is an example:

Let $(\phi,\theta)$ be geographical coordinates with $\theta=\pm{\pi\over2}$ at the poles, and define $$f:\quad (\phi,\theta)\mapsto f(\phi,\theta):=\bigl(\phi,{\pi\over2}\sin\theta\bigr)\ .$$

Möbius transformations are studied in detail in any elementary complex analysis text. Homeomorphisms of spheres are studied in (algebraic) topology. The case of $S^1$ plays a big rôle in dynamical systems (rotation number).