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Let $E \to B$ be a topological fiber bundle. We know that the Serre spectral sequence allows you to compute the cohomology of $E$ with constant coefficient in terms of the cohomology of $F$ and the cohomology of $B$ (with non constant coefficients).

Is there an analog of this to compute the cohomology of $E$ with non constant coefficients (say with a locally constant system of coefficients)?

I'm interested in the general answer, but let me add that I'm trying to do computations in a case where both the fibre and the base are smooth manifolds. Even more precisely, the fibre is a compact Lie group.

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    $\begingroup$ Yes, see en.wikipedia.org/wiki/Leray_spectral_sequence $\endgroup$
    – SashaP
    Commented Sep 29, 2016 at 9:48
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    $\begingroup$ For a fiber bundle you probably don't need the full Leray spectral sequence. You just want to know that for a local coefficient system $\mathcal{A}$ on $E$, there is a spectral sequence starting from $H^p(B;H^q(F;\mathcal{A}|_F))$ and converging to $H^*(E;\mathcal{A})$. This is true; I could probably dig out a reference if this is what you are asking for. $\endgroup$
    – Mark Grant
    Commented Sep 29, 2016 at 14:57
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    $\begingroup$ Hi, thank you very much to you both. Dear Mark, this is exactly what I was hoping for and I would be very much interested in a reference, thanks. $\endgroup$
    – cannonball
    Commented Sep 29, 2016 at 16:04

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