 Fixed-point property of even-dimensional complex projective space

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.