Cup product

Summary

The cup product is a multiplication $H^k(X; R) \times H^l(X; R) \to H^{k+l}(X; R)$ on singular cohomology with coefficients in a commutative ring $R$. It is associative, has a unit, and is graded commutative. It makes cohomology into a ring, the cohomology ring, which carries much more information than the cohomology groups alone.