Paper 3, Section II, D

Define the cross-ratio $\left[a_{0}, a_{1}, a_{2}, z\right]$ of four points $a_{0}, a_{1}, a_{2}, z$ in $\mathbb{C} \cup\{\infty\}$, with $a_{0}, a_{1}, a_{2}$ distinct.

Let $a_{0}, a_{1}, a_{2}$ be three distinct points. Show that, for every value $w \in \mathbb{C} \cup\{\infty\}$, there is a unique point $z \in \mathbb{C} \cup\{\infty\}$ with $\left[a_{0}, a_{1}, a_{2}, z\right]=w$. Let $S$ be the set of points $z$ for which the cross-ratio $\left[a_{0}, a_{1}, a_{2}, z\right]$ is in $\mathbb{R} \cup\{\infty\}$. Show that $S$ is either a circle or else a straight line together with $\infty$.

A map $J: \mathbb{C} \cup\{\infty\} \rightarrow \mathbb{C} \cup\{\infty\}$ satisfies

$\left[a_{0}, a_{1}, a_{2}, J(z)\right]=\overline{\left[a_{0}, a_{1}, a_{2}, z\right]}$

for each value of $z$. Show that this gives a well-defined map $J$ with $J^{2}$ equal to the identity.

When the three points $a_{0}, a_{1}, a_{2}$ all lie on the real line, show that $J$ must be the conjugation map $J: z \mapsto \bar{z}$. Deduce from this that, for any three distinct points $a_{0}, a_{1}, a_{2}$, the map $J$ depends only on the circle (or straight line) through $a_{0}, a_{1}, a_{2}$ and not on their particular values.