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$\DeclareMathOperator\BPSO{BPSO}\DeclareMathOperator\PSO{PSO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\BSO{BSO}$The projective group is $\PSO(2d)=\SO(2d)/Z_2$.

$\BPSO(2d)$ is the classifying space of $\PSO(2d)$.

Question: What is the cohomology of $\BPSO(2d)$ with $Z_2$ coefficients?

Note: Here we already know the cohomology of $\BSO(2d)$ with $Z_2$ coefficients. $$H^*(\BSO(2d),Z_2)=Z_2[w_2,w_3,...,w_{2d}].$$ Can we obtain that of $\BPSO(2d)$?

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    $\begingroup$ See Toda's "Cohomology of Classifying Spaces" Proposition 4.5 for the case of odd $d$. $\endgroup$ Sep 12, 2021 at 16:49
  • $\begingroup$ @AleksandarMilivojevic Thanks! I had a look. Could you write it as an answer too? $\endgroup$
    – user34104
    Sep 13, 2021 at 12:35

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