Lefschetz fixed-point property of even-dimensional real projective space ($L(f)\equiv 1(\mathrm{mod}2)$)

Every self-map has $f^{\ast }a=\lambda a$ 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.