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I consider the following two situations:

  1. Let $B$ be a simply connected space, and $F\to E\to B$, $F'\to E'\to B$ two fibrations with a map $f:E'\to E$ sending fibers to fibers and inducing the identity on $B$ (in my case $f$ is injective but I don't think it matters).

  2. Let $f:B\to B'$ be a map between simply connected spaces, $F\to E\to B$ a fibration, and $F\to E'\to B'$ the pullback with respect to $f$. Here as well, I deal with $f$ being injective.

Question: is there some relative version of the Serre spectral sequence that allows me to compute the relative cohomology groups of $(E,E')$ from the cohomology of $(F,F')$ and $B$ (case 1), resp. the cohomology of $F$ and $(B,B')$ (case 2)?

Some reference would be perfect. If some topological assumption on the spaces or maps are required, I am dealing with classifying spaces and Borel constructions applied to smooth actions on manifolds, so it is very well behaved stuff.

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2 Answers 2

up vote 6 down vote accepted

These relative versions of the Leray-Serre spectral sequence appear as Exercises 5.5 and 5.6 in McCleary's "A User's Guide to Spectral Sequences". Your case 2 is also mentioned on p17 of Allen Hatcher's "Spectral Sequences in Algebraic Topology" book project, available here.

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Thank you very much for the references! It's exactly what I was looking for. –  Marco Radeschi Mar 23 at 16:45

I won't give you a reference as most people use it as a common place. Probably, a good textbook in algebraic topology. (And for a proof, one can just refer to the Leray spectral sequence.) Of course, everything works, both ways. Given a bundle $F\to E\to B$, a subbundle $F'\to E'\to B$, and a subspace $B'\subset B$, there is a spectral sequence $$H^p(B,B';H^q(F,F'))\Rightarrow H^{p+q}(E,E'\cup E|_{B'}).$$ If things are not simply connected, one should use local systems of coefficients.

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