Cellular homology of real projective space

Real projective space has one cell in every dimension from zero through $n$. With integral coefficients its cellular differential $C_k\to C_{k-1}$ is multiplication by $1+(-1{)}^k$, hence is zero for odd $k$ and multiplication by two for even $k$. With coefficients in ${\mathbb{F}}_2$, every cellular differential vanishes.