2
$\begingroup$

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

$\endgroup$

1 Answer 1

4
$\begingroup$
  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 $\deg\alpha = \deg\beta = 1$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.