Cohomology ring of classifying space of spin group $\mathrm{BSpin}(n)$

Question

$\DeclareMathOperator\BSpin{BSpin}\DeclareMathOperator\Pin{Pin}\DeclareMathOperator\BSO{BSO}\DeclareMathOperator\BO{BO}\DeclareMathOperator\BPin{BPin}$In the answer for question: Homology of classifying space of spin group BSpin(n), it was shown that $H_i(\BSpin(\infty),\mathbb{Z})$ is $0,0,0,\mathbb{Z}$, for $i=1,2,3,4$. Here I like ask a more detailed question:

• What is the cohomology ring $H^*(\BSpin(\infty),Z)$?

Also what is cohomology rings $H^i(\BPin^\pm(\infty),Z)$ (where $\Pin^\pm (n)$ is a $\Pin^\pm$ group)?

This paper

• D.J. Benson and Jay A. Wood, Integral invariants and cohomology of $\BSpin(n)$, Topology 34 Issue 1 (1995) pp 13–28, doi:10.1016/0040-9383(94)E0019-G,

does not give an explicit result.

I cannot find a digital copy of

• E. Thomas, On the cohomology groups of the classifying space for the stable spinor group, Bol. Sot. Mat. Mexicana (2) 7 (1962), 57-69.

For $\BSO(n)$, this paper

• Edgar H. Brown, Jr., The Cohomology of $\BSO_n$ and $\BO_n$ with Integer Coefficients, Proceedings of the American Mathematical Society Vol. 85, No. 2 (1982), pp. 283-288, doi:10.2307/2044298,

provides a full answer.

Answer 1

• Haibao Duan, The characteristic classes and Weyl invariants of Spinor groups, https://arxiv.org/abs/1810.03799

This recent paper gives a clean answer for your question. In particular, it provides the so-called spin classes, with explicit relations to Pontryagin classes and Stiefel-Whiteny classes