# Proof of Hopf's theorem (the one about cohomology of Lie groups being equal to the cohomology of a product of spheres)

Theorem (H. Hopf). Let $G$ be a compact connected Lie group. Then $G$ has the real cohomology of a product of odd dimensional spheres, $H(G,\mathbb{R}) \approx H(\prod_q S^{2k_q - 1},\mathbb{R})$.

The only place I could find the proof of this theorem was the book by Greub, Connections, curvature and cohomology. Do you guys know where else can I find a proof of this theorem? The original papers are in german and french, and I couldn't find any translations. Thanks in advance.

## 1 Answer

This is a corollary of Theorem 3C.4 in Hatcher's Algebraic Topology:

If $A$ is a [connected graded] commutative, associative Hopf algebra over a field $F$ of characteristic 0, and $A^n$ is finite-dimensional over $F$ for each $n$, then $A$ is isomorphic as an algebra to the tensor product of an exterior algebra on odd-dimensional generators and a polynomial algebra on even-dimensional generators.

The cohomology of a compact connected Lie group is a Hopf algebra satisfying these conditions, and additionally is $0$ in all but finitely degrees. This means that there cannot be any even-dimensional generators, so you have an exterior algebra on odd-dimensional generators, which is isomorphic to the cohomology ring of $\prod_q S^{2k_q-1}$ where the numbers $2k_q-1$ are the dimensions of the generators.