= Fixed-point property of even-dimensional complex projective space
{title2=$L(f)=1+d+\cdots+d^{2r}\ne0$}
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 $d\ne1$ 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.
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