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Let $F\rightarrow X\rightarrow B$ be a fibration. If we know very well the spaces $F$ and $B$ and wish to compute the homology of $X$. One possible tool is the Serre Spectral Sequence. However, it works under the condition that $\pi_1(B)$ acts trivially on $H_*(F;G)$. If this condition ($\pi_1(B)$ acts trivially on $H_*(F;G)$) does not hold, what other tools can one use to compute the homology of the homology of $X$?

In fact I am interested in the special case that all spaces in the fibration are $K(\pi,1)$ spaces. If any approach works for this particular case it would be wonderful.

Thank you!

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    $\begingroup$ I believe you can still use the Serre spectral sequence but you need to use local coefficients. $\endgroup$
    – Callan McGill
    Commented Feb 15, 2012 at 3:10
  • $\begingroup$ As Callan mentions, there is a Serre spectral sequence for any fibration (and a Leray spectral sequence for any map). It's a little more complicated technically due to the local coefficients but it's perfectly usable. For this Serre spectral sequence, it's not enough to know $H_*(B)$ and $H_*(F)$ independently, but you have to know $H_*(B; H_*(F))$, homology in the twisted system. $\endgroup$
    – Ryan Budney
    Commented Feb 15, 2012 at 3:48
  • $\begingroup$ @Ryan, May I know where I can find any reference on the Serre Spectral sequence for any fibration mentioned in your comment? Thank you! $\endgroup$
    – Zuriel
    Commented Feb 15, 2012 at 3:59
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    $\begingroup$ In the case where all spaces are $K(\pi,1)$'s this is called the Lyndon-Hochschild-Serre SS and can be found in any book on group (co)homology. Knowing this doesn't aid your computations though... $\endgroup$
    – Mark Grant
    Commented Feb 15, 2012 at 7:59
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    $\begingroup$ There's a nice treatment of the general case in Davis-Kirk Lectures Notes in Algebraic Topology, as well as a chapter on local coefficients if you need brushing up on that. $\endgroup$
    – Greg Friedman
    Commented Feb 15, 2012 at 8:25

1 Answer 1

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Repeating Mark Grant's comment, the spectral sequence when all spaces are $K(\pi,1)$s goes under the name Lyndon-Hochschild-Serre spectral sequence.

Good references for this spectral sequence are:

D. Benson: Representations and Cohomology II: cohomology of groups and modules

L. Evens: The cohomology of groups

Loads of papers have been written about this spectral sequence: Calculating $E^2$, when it degenerates at $E^2$, differentials, extension problems, you name it.... But a lot of the details depend on which class of groups $\pi$ you are interested in, so it's hard to give specific pointers without more information.

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