---
tag: WJXK0V9NE9
kind: concept
name: Hopf algebra structure on cohomology
msc: [55P45, 16T05, 57T05]
sources:
  - url: https://en.wikipedia.org/wiki/Hopf_algebra
---

## Summary

The cohomology with field coefficients of a connected [[H-space|EASHTX8EG5]] 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.

## Explanations

### L1 human

On an H-space one can multiply points, and this multiplication induces a map in cohomology that goes the other way, from the cohomology of $X$ to that of $X \times X$.
So the cohomology gets both a product, the cup product, and a "coproduct".
The two operations are compatible, which is the definition of a Hopf algebra.
For the $3$-sphere, seen as the unit quaternions, the cohomology is spanned by $1$ and a class $x$ of degree $3$, and the coproduct sends $x$ to $x \otimes 1 + 1 \otimes x$.