Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!

share|improve this question
5  
I believe you can still use the Serre spectral sequence but you need to use local coefficients. –  Callan McGill Feb 15 '12 at 3:10
    
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. –  Ryan Budney Feb 15 '12 at 3:48
    
@Ryan, May I know where I can find any reference on the Serre Spectral sequence for any fibration mentioned in your comment? Thank you! –  Zuriel Feb 15 '12 at 3:59
2  
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... –  Mark Grant Feb 15 '12 at 7:59
1  
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. –  Greg Friedman Feb 15 '12 at 8:25
show 1 more comment

1 Answer

up vote 7 down vote accepted

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.