The cup product is a product on cohomology classes. In the case of de Rham cohomology, a cohomology class can be represented by a closed form. The cup product of [α] and [β] is represented by the closed form [α⋀β], where ⋀ is the wedge product of differential forms. It is the dual operation to intersection in homology. In general, the cup product is a map ⋁:H^p×H^q->H^(p + q) which satisfies a⋁b = (-1)^(p q) b⋁a, where H^k is the kth cohomology group.

cup | de Rham cohomology | homology