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  "body": "\n## Summary\n\nThe cohomology with field coefficients of a connected [[H-space|EASHTX8EG5]] of finite type is a Hopf algebra.\nThe cup product is its multiplication, and the map induced by the H-space multiplication is its comultiplication.\nHopf discovered this structure in 1941.\nHe used it to show that the rational cohomology of a compact connected Lie group is an exterior algebra on generators of odd degree.\nThe structure theory of Hopf algebras, due to Borel, Milnor and Moore, then restricts which spaces can be H-spaces.\n\n## Explanations\n\n### L1 human\n\nOn 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 $X \\times X$.\nSo the cohomology gets both a product, the cup product, and a \"coproduct\".\nThe two operations are compatible, which is the definition of a Hopf algebra.\nFor 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 $x \\otimes 1 + 1 \\otimes x$.\n",
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