de Rham cohomology of $U(2)$

Question

I'm trying to calculate the de Rham cohomology of $U(2)$, but I don't know how to do this. I'd like to avoid Mayer-Vietoris if possible. I'm doing this in preparation for an exam in my topology course next week, so any help would be appreciated.

Answer 1

Here are two facts that you can use to compute the cohomology of $U(2)$:

• $U(n)$ is diffeomorphic to $SU(n)\times S^1$; see this answer for example.
• $SU(2)$ is diffeomorphic to $S^3$, see here for example.

Therefore $U(2)$ is diffeomorphic to $S^3\times S^1$. It follows from the Künneth Theorem that $H_{\text{dR}}^*(U(2)) \cong \mathbb{R}[\alpha, \beta]/(\alpha^2, \beta^2)$ where $\deg\alpha = 3$ and $\deg\beta = 1$.