Geodesic
A Note on the mod 2 Cohomology of BŜOn 〈l6〉
Canadian journal of mathematics, Tome 37 (1985) no. 5, pp. 893-907

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Recall BSOn is the classifying space for the special orthogonal group of rank n. It is well known that the mod 2 cohomology ring of BSOn is given as follows: where wi is the i-th mod 2 Stiefel-Whitney class. For the 3-connective cover of BSOn , BSpinn , Quillen in [4] has determined H*(BSpinn ) completely for all n. Let BŜOn 〈8〉, BŜOn 〈16〉 be the classifying spaces for n-plane spin bundle ξ satisfying w 4(ξ) = 0 and w 4(ξ) = w 8(ξ) = 0 respectively. This note follows the method of A. Borel [2] and gives the mod 2 cohomology ring of BŜOn 〈8〉 and BŜOn 〈16〉 for small n. In particular we answer the question “when is H*(BŜOn 〈8〉; Z 2), or H*(BŜOn 〈16〉; Z 2) a polynomial algebra?” 
Ng, Tze-Beng. A Note on the mod 2 Cohomology of BŜOn 〈l6〉. Canadian journal of mathematics, Tome 37 (1985) no. 5, pp. 893-907. doi: 10.4153/CJM-1985-048-5
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