TITLE: Homeomorphism with finitely many fixed points acting by -1 on $H^2(CP^2)$
QUESTION [7 upvotes]: Is there a homeomorphism $f:CP^2\to CP^2$ that has finitely many fixed points and acts by -1 on $H^2$?

REPLY [12 votes]: 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.