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Given an oriented $S^k$ bundle $E$ over a compact manifold $M$ we get the Gysin Sequence (I am interested in the DeRham cohomology). We can obtain this sequence from the Serre Spectral Sequence if the base is simply connected, but the Gysin Sequence exists even for non-simply connected base manifolds via the Thom Isomorphism.

My question is, suppose instead of $S^k$ we have some other fiber $F$ bundle, of interest to me is a principal $S^1 \times \cdots \times S^1$ bundle, then if the base manifold is simply connected we get the Serre Spectral Sequence, but if it is not does there still exist some possibly weaker spectral sequence like the Gysin sequence but with non twisted coefficients, again for DeRham cohomology.

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