Fixed-point property of even-dimensional complex projective space ($L(f)=1+d+\cdots +d^{2r}\ne 0$)

Every continuous self-map of ${\mathbb{CP}}^{2r}$ has a fixed point. If its pullback sends the integral degree-two generator to $d$ times itself, multiplicativity gives trace $d^j$ in degree $2j$. Thus its Lefschetz number is $1+d+\cdots +d^{2r}$, which is positive: for $d\ne 1$ it is $(1-d^{2r+1})/(1-d)>0$, and at $d=1$ it is $2r+1$. The Lefschetz fixed-point theorem applies. In particular no nontrivial finite group acts freely on this simply connected space.

The Complex projective plane is simply connected. If it were the total space of a covering of degree at least two, it would be the universal cover and have a nonidentity deck transformation. Such a transformation acts by $u\mapsto \pm u$ on $H^{\ast }({\mathbb{CP}}^2)=\mathbb{Z}[u]/(u^3)$, giving Lefschetz number 1 or 3. It therefore has a fixed point, impossible for a nonidentity deck transformation.