Cohomology of classifying space

Question

I would like to know if anyone knows how to calculate the cohomology of the following spaces, especially in the case of classifying spaces:

1) $ H^\ast (BSU(2), \mathbb{Z}) $

2) $ H^\ast (BO(3), \mathbb{Z}/2) $

3) $ H^\ast (O(3), \mathbb{Z}/2) $

Thank you

Answer 1

1. Note that $BSU(2) = BSp(1) = \mathbb{HP}^{\infty}$, so $H^*(BSU(2); \mathbb{Z}) \cong \mathbb{Z}[\alpha]$ where $\deg\alpha = 4$.

2. In general, $H^*(BO(n); \mathbb{Z}_2) \cong \mathbb{Z}_2[w_1, \dots, w_n]$ where $\deg w_i = i$.

3. Note that $O(3) \cong SO(3)\times\mathbb{Z}_2$ and $SO(3)$ is diffeomorphic to $\mathbb{RP}^3$, so $O(3)$ is diffeomorphic to $\mathbb{RP}^3\sqcup\mathbb{RP}^3$. Therefore $H^*(O(3); \mathbb{Z}_2) \cong \mathbb{Z}_2[\alpha, \beta]/(\alpha\beta, \alpha^4, \beta^4)$ where $\deg\alpha = \deg\beta = 1$.