It is well known that if $G$ is a lie group and $H$ is a closed subgroup of $G$, the inclusion $H \hookrightarrow G$ induces a fiber bundle on the classifying spaces

$$ G/H \rightarrow BH \rightarrow BG$$

I am interested in the cohomological Serre spectral sequence (with coefficients in $\mathbb{Z}/2\mathbb{Z}$ for simplicity) associated to this fibration; namely, the $E_2$ term is given by

$$E_2^{p,q} = H^p(BG;\mathcal{H}^q(G/H))$$

If $G$ is path-connected, $BG$ is simply connected and therefore $\mathcal{H}^q(G/H)$ is thus the usual cohomology $H^q(G/H)$; However, what can I say in the case of $G$ being not path-connected? Is there any example where the twisted coefficients are not trivial? or they are trivial in my setting.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.