Cohomology ring

Specifically, given a sequence of cohomology groups H^k(X;R) on X with coefficients in a commutative ring R (typically R is Z_n, Z, Q, R, or C) one can define the cup product, which takes the form

${\displaystyle H^k(X;R)\times H^{\ell }(X;R)\to H^{k+\ell }(X;R).}$

The cup product gives a multiplication on the direct sum of the cohomology groups

${\displaystyle H^{\bullet }(X;R)=\underset{k\in \mathbb{N}}{⨁}{H}^k(X;R).}$

This multiplication turns H^•(X;R) into a ring. In fact, it is naturally an N-graded ring with the nonnegative integer k serving as the degree. The cup product respects this grading.

The cohomology ring is graded-commutative in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degree k and ℓ; we have

${\displaystyle ({\alpha }^k⌣{\beta }^{\ell })=(-1{)}^{k\ell }({\beta }^{\ell }⌣{\alpha }^k).}$

A numerical invariant derived from the cohomology ring is the cup-length, which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example, a complex projective space has cup-length equal to its complex dimension.

Examples

In what follows, vertical bars around an element denote its dimension in the cohomology ring.

• ${\displaystyle {\mathrm{H}}^{\ast }(\mathbb{R}{P}^n;{\mathbb{F}}_2)={\mathbb{F}}_2[\alpha ]/({\alpha }^{n+1})}$ where ${\displaystyle |\alpha |=1}$.
• ${\displaystyle {\mathrm{H}}^{\ast }(\mathbb{R}{P}^{\mathrm{\infty }};{\mathbb{F}}_2)={\mathbb{F}}_2[\alpha ]}$ where ${\displaystyle |\alpha |=1}$.
• ${\displaystyle {\mathrm{H}}^{\ast }(\mathbb{C}{P}^n;\mathbb{Z})=\mathbb{Z}[\alpha ]/({\alpha }^{n+1})}$ where ${\displaystyle |\alpha |=2}$.
• ${\displaystyle {\mathrm{H}}^{\ast }(\mathbb{C}{P}^{\mathrm{\infty }};\mathbb{Z})=\mathbb{Z}[\alpha ]}$ where ${\displaystyle |\alpha |=2}$.
• ${\displaystyle {\mathrm{H}}^{\ast }(\mathbb{H}{P}^n;\mathbb{Z})=\mathbb{Z}[\alpha ]/({\alpha }^{n+1})}$ where ${\displaystyle |\alpha |=4}$.
• ${\displaystyle {\mathrm{H}}^{\ast }(\mathbb{H}{P}^{\mathrm{\infty }};\mathbb{Z})=\mathbb{Z}[\alpha ]}$ where ${\displaystyle |\alpha |=4}$.
• ${\displaystyle {\mathrm{H}}^{\ast }(T^2;\mathbb{Z})={\mathrm{\Lambda }}_{\mathbb{Z}}[{\alpha }_1,{\alpha }_2]}$ where ${\displaystyle |{\alpha }_1|=|{\alpha }_2|=1}$.
• ${\displaystyle {\mathrm{H}}^{\ast }(T^n;\mathbb{Z})={\mathrm{\Lambda }}_{\mathbb{Z}}[{\alpha }_1,...,{\alpha }_n]}$ where ${\displaystyle |{\alpha }_i|=1}$.
• ${\displaystyle {\mathrm{H}}^{\ast }(S^n;\mathbb{Z})=\mathbb{Z}[\alpha ]/[{\alpha }^2]}$ where ${\displaystyle |\alpha |=n}$.
• If ${\displaystyle K}$ is the Klein bottle, ${\displaystyle {\mathrm{H}}^{\ast }(K;\mathbb{Z})=\mathbb{Z}[\alpha ,\beta ]/[{\alpha }^2,2\beta ,\alpha \beta ,{\beta }^2]}$ where ${\displaystyle |\alpha |=1,|\beta |=2}$.
• By the Künneth formula, the mod 2 cohomology ring of the cartesian product of n copies of ${\displaystyle \mathbb{R}{P}^{\mathrm{\infty }}}$ is a polynomial ring in n variables with coefficients in ${\displaystyle {\mathbb{F}}_2}$.
• 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.