Let $G$ be a group. Let a principal $G$-bundle $G\to E\to B$. Then we have a fiber sequence $G\to E\to B\to BG$.

Let $k$ be a field. Suppose $H^*(BG;k)$ and $H^*(E,k)$ are known. How to get $H^*(B;k)$?

The action of $\pi_1(BG)$ may not act trivially on $H^*(E;k)$. Hence in Serre spectral sequence, local coefficients need to be used. However, local coefficients cohomology is unknown because we are not clear about the action of $\pi_1(BG)$ on $H^*(E,k)$. What should I do?

  • $\begingroup$ I'm a bit rusty on my algebraic topology, but doesn't the Leray-Hirsch theorem apply? $\endgroup$ – Matt Samuel Jan 17 '15 at 2:30

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