Inmathematics, specificallyalgebraic topology, thecohomology ring of atopological spaceX is aring formed from thecohomology groups ofX together with thecup product serving as the ring multiplication. Here 'cohomology' is usually understood assingular cohomology, but the ring structure is also present in other theories such asde Rham cohomology. It is alsofunctorial: for acontinuous mapping of spaces one obtains aring homomorphism on cohomology rings, which is contravariant.

Specifically, given a sequence of cohomology groupsH^k(X;R) onX with coefficients in acommutative ringR (typicallyR isZ_n,Z,Q,R, orC) one can define thecup 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 thedirect sum of the cohomology groups

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

This multiplication turnsH^•(X;R) into a ring. In fact, it is naturally anN-graded ring with the nonnegative integerk serving as the degree. The cup product respects this grading.

The cohomology ring isgraded-commutative in the sense that the cup product commutes up to a sign determined by the grading. Specifically, for pure elements of degreek 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 thecup-length, which means the maximum number of graded elements of degree ≥ 1 that when multiplied give a non-zero result. For example, acomplex projective space has cup-length equal to itscomplex dimension.

• ${\displaystyle {\mathrm{H}}^{\ast }(\mathbb{R}{P}^n;{\mathbb{F}}_2)={\mathbb{F}}_2[\alpha ]/({\alpha }^{n+1})}$  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 theKünneth formula, the mod 2 cohomology ring of the cartesian product ofn copies of${\displaystyle \mathbb{R}{P}^{\mathrm{\infty }}}$  is a polynomial ring inn 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 ofsuspensions vanishes except for the degree 0 part.

• Quantum cohomology

• Novikov, S. P. (1996).Topology I, General Survey. Springer-Verlag.ISBN 7-03-016673-6.
• Hatcher, Allen (2002),Algebraic Topology, Cambridge: Cambridge University Press,ISBN 0-521-79540-0.