Hopf algebra structure on cohomology

Summary

The cohomology with field coefficients of a connected H-space of finite type is a Hopf algebra. The cup product is its multiplication, and the map induced by the H-space multiplication is its comultiplication. Hopf discovered this structure in 1941. He used it to show that the rational cohomology of a compact connected Lie group is an exterior algebra on generators of odd degree. The structure theory of Hopf algebras, due to Borel, Milnor and Moore, then restricts which spaces can be H-spaces.