Cohomology in groups

Question

I'm trying to find the Rham cohomology of the groups $SU(2)$ and $U(2)$. I know that $SU(2)$ is isomorphic to $S^3$ but I don't know what is $U(2)$ isomorphic to. My question is: if $SU(2) \simeq S^3$ and I know the Rham cohomology of $S^3$, can I say that that cohomology is the one for $SU(2)$? If not, how can I compute it?

I'm really stucked on this problem and any help would be great.

Thanks in advance!

Answer 1

The de Rham cohomology of a Lie group $G$ is the de Rham cohomology of the underlying manifold. So the de Rham cohomology of $SU(2)$ is simply that of $S^3$, as you guess. As a manifold, we have

$$ U(n) = SU(n) \times S^1 $$

although one should beware that this is not true considered as an isomorphism of groups.

Note that a cohomology group of a product manifold can be obtained from those of the manifolds in the product according to the Künneth formula.