{title2=$H^*(\mathbb{RP}^n;\mathbb Z)$}

For $m\geq0$, the <cohomology ring> is
$$
H^*(\mathbb{RP}^{2m};\mathbb Z)\cong\mathbb Z[a]/(2a,a^{m+1}),\qquad |a|=2,
$$
and
$$
H^*(\mathbb{RP}^{2m+1};\mathbb Z)\cong\mathbb Z[a,b]/(2a,a^{m+1},ab,b^2),\qquad |a|=2,\quad |b|=2m+1.
$$
The generator $a$ reduces to the square of the degree-one generator in the <mod-two cohomology ring of real projective space>; $b$ is the integral <orientation class> in odd top dimension.


 Integral cohomology ring of real projective space

ID: integral-cohomology-ring-of-real-projective-space