Is there a homeomorphism $f:CP^2\to CP^2$ that has finitely many fixed points and acts by -1 on $H^2$?



Homeomorphism with finitely many fixed points acting by -1 on $H^2(CP^2)$
Let $a\_1$, $a\_2$ and $a\_3$ be three distinct positive real numbers. Take the map
$$[z\_1 : z\_2 : z\_3] \mapsto [a\_1 \overline{z\_1}: a\_2 \overline{z\_2} : a\_3 \overline{z\_3}]$$
where the bar is complex conjugation. I claim that the only fixed points are $[1:0:0]$, $[0:1:0]$ and $[0:0:1]$.


Suppose that to the contrary that $[z\_1:z\_2:z\_3]$ is fixed and that $z\_i z\_j \neq 0$ for some indices $i \neq j$. Then there is a scalar $\lambda$ such that
$$\lambda = \frac{a\_i \overline{z\_i}}{z\_i} = \frac{a\_j \overline{z\_j}}{z\_j}.$$
But then we have
$$|\lambda| = a\_i = a\_j,$$
contradicting that the $a$'s are distinct.


