In what follows, vertical bars around an element denote its dimension in the cohomology ring. • \operatorname{H}^*(\mathbb{R}P^n; \mathbb{F}_2) = \mathbb{F}_2[\alpha]/(\alpha^{n+1}) where |\alpha|=1. • \operatorname{H}^*(\mathbb{R}P^\infty; \mathbb{F}_2) = \mathbb{F}_2[\alpha] where |\alpha|=1. • \operatorname{H}^*(\mathbb{C}P^n; \mathbb{Z}) = \mathbb{Z}[\alpha]/(\alpha^{n+1}) where |\alpha|=2. • \operatorname{H}^*(\mathbb{C}P^\infty; \mathbb{Z}) = \mathbb{Z}[\alpha] where |\alpha|=2. • \operatorname{H}^*(\mathbb{H}P^n; \mathbb{Z}) = \mathbb{Z}[\alpha]/(\alpha^{n+1}) where |\alpha|=4. • \operatorname{H}^*(\mathbb{H}P^\infty; \mathbb{Z}) = \mathbb{Z}[\alpha] where |\alpha|=4. • \operatorname{H}^*(T^2;\mathbb{Z})=\Lambda_\mathbb{Z}[\alpha_1, \alpha_2] where |\alpha_1|=|\alpha_2|=1. • \operatorname{H}^*(T^n;\mathbb{Z})=\Lambda_\mathbb{Z}[\alpha_1,..., \alpha_n] where |\alpha_i|=1. • \operatorname{H}^*(S^n;\mathbb{Z})= \mathbb{Z}[\alpha]/[\alpha^2] where |\alpha|=n . • If K is the Klein bottle, \operatorname{H}^*(K;\mathbb{Z})= \mathbb{Z}[\alpha,\beta]/[\alpha^2,2\beta,\alpha\beta,\beta^2] where |\alpha|=1, |\beta|=2. • By the
Künneth formula, the mod 2 cohomology ring of the cartesian product of
n copies of \mathbb{R}P^\infty is a polynomial ring in
n variables with coefficients in \mathbb{F}_2. • The reduced cohomology ring of wedge sums is the direct product of their reduced cohomology rings. • The cohomology ring of
suspensions vanishes except for the degree 0 part. == See also ==