# Cohomology of $\mathrm{BPSO}(2d)$ with $Z_2$ coefficients

$$\DeclareMathOperator\BPSO{BPSO}\DeclareMathOperator\PSO{PSO}\DeclareMathOperator\SO{SO}\DeclareMathOperator\BSO{BSO}$$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 $$\BPSO(2d)$$?

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