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I have added statement of the integral cohomology ring of $B SO(3)$ (thanks to David R. for discussion), following Brown 82, or I hope I did:
$H^\bullet\big( B SO(3), \mathbb{Z} \big) \;\simeq\; \mathbb{Z}\big[ p_1, W_3\big] / (2 W_3) \,,$Brown’s Theorem 1.5 has a last tedious clause with a relation satisfied by the cup product of two SW classes. It looks to me that the whole clause collapses to nothing in the present case. But needs to be double checked.
Here’s another source
Mark Feshbach, The Integral Cohomology Rings of the Classifying Spaces of O(n) and SO(n), Indiana Univ. Math. J. 32 (1983), 511-516. doi:10.1512/iumj.1983.32.32036
The generators are indeed as you say. The relations are complicated, but I think they all turn out to be (except for $2W_3=0$) vacuous, for $n=3$… I think.
Here is another source, that looks a bit more tractable. If that Google Books link is not readable, it’s Section 4.2 of Differential Geometry and Mathematical Physics: Part II. Fibre Bundles, Topology and Gauge Fields by Gerd Rudolph and Matthias Schmidt (doi:10.1007/978-94-024-0959-8)
Edit: Hmm, it looks to be open access. Try this link to go to the section in HTML format.
Remark 4.2.25.1 of Rudolph–Schmidt’s book, together with Theorem 4.2.23 and Corollary 4.2.24 I think close the deal. It seems $2W_3=0$ is the only relation, so that the cohomology ring is $\mathbb{Z}[p_1,W_3]/(2W_3)$.
Thanks again!
Yes, that book Rudolph-Schmidt shows some more effort to polish-up the statement of the theorem.
It seems to be a nice book, generally. Have added pointer to it to a bunch of nLab pages now.
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