Lefschetz fixed-point property of even-dimensional real projective space
= Lefschetz fixed-point property of even-dimensional real projective space
{c}
{title2=$L(f)\equiv1\pmod2$}
Every self-map has $f^*a=\lambda a$ in the mod-two cohomology ring, with $\lambda\in\mathbb F_2$. Its mod-two Lefschetz number is $1+\lambda+\cdots+\lambda^{2r}$, equal to one for either value of $\lambda$. This is the reduction of the integral cellular Lefschetz number, so the latter is nonzero and a fixed point exists.