Products in Homology and Cohomology

Abstract

The most important product is undoubtedly the so-called cup product: It assigns to any elements u ∈ H^p(X; G₁) and v ∈ H^q(X; G₂ an element u ∪ v ∈ H^p+q(X; G₁ ⊗ G₂). This product is bilinear (or distributive) and is natural with respect to homomorphisms induced by continuous maps. It is an additional element of structure on the cohomology groups that often allows one to distinguish between spaces of different homotopy types, even though they have isomorphic homology and cohomology groups. This additional structure also imposes restrictions on the possible homomorphisms which can be induced by continuous maps.