Computing the homology groups of spaces in a fibration - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:00:48Z http://mathoverflow.net/feeds/question/88481 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88481/computing-the-homology-groups-of-spaces-in-a-fibration Computing the homology groups of spaces in a fibration Zuriel 2012-02-15T02:50:52Z 2012-02-15T09:03:27Z <p>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 $X$?</p> <p>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.</p> <p>Thank you! </p> http://mathoverflow.net/questions/88481/computing-the-homology-groups-of-spaces-in-a-fibration/88502#88502 Answer by Jesper Grodal for Computing the homology groups of spaces in a fibration Jesper Grodal 2012-02-15T09:03:27Z 2012-02-15T09:03:27Z <p>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. </p> <p>Good references for this spectral sequence are:</p> <p>D. Benson: Representations and Cohomology II: cohomology of groups and modules</p> <p>L. Evens: The cohomology of groups</p> <p>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.</p>