Is the Serre spectral sequence a special case of the Leray spectral sequence?

Let FEB be a fibration with B simply connected (more generally, such that π1(B) acts trivially on the homology of F). Then there is a Serre spectral sequence Hp(B,Hq(F))Hp+q(E). One can do the same for singular cohomology. However, for reasonable spaces (specifically, locally contractible spaces, e.g. CW complexes), singular cohomology is the same as sheaf cohomology of the constant sheaf Z.

But there is another spectral sequence for sheaf cohomology: the Leray spectral sequence. Given spaces X,Y and f:XY, and a sheaf F on X, there is a spectral sequence Hp(Y,Rqf(F))Hp+q(X,F).
The Wikipedia article hints that the topological implications of this include in particular the Serre spectral sequence. I would be interested in this, because I like the machinery of the Grothendieck spectral sequence (from which the Leray spectral sequence easily follows), and would be curious if the Serre spectral sequence could be obtained as a corollary.

Is this possible?


Yes. In fact, the result is basically obvious if you use Czech cohomology on the base.

Serre really had two key insights. First, sheaf cohomology is a pain to compute, but if there is no fundamental group then for fiber bundles the Leray spectral sequence is really just using normal old-fashioned untwisted cohomology. Second, you don’t really need to work with fiber bundles — all you need are Serre fibrations, and those are easy to construct. In particular, you have the standard Serre fibration ΩXPXX, where ΩX is the loop space of X and PX is the space of paths starting at the basepoint of X and the map PXX is “evaluation at the endpoint”. Clearly PX is contractible! An amazing amount of milage can be had from this silly observation!

Serre also really developed many of the key algebraic tricks one needs to work with spectral sequences. For instance, he had the amazing idea that one can work modulo “Serre classes”, and thus ignore things like torsion. It’s like pretending to localize spaces long before Sullivan and Quillen realized you could do so for real!

Source : Link , Question Author : Akhil Mathew , Answer Author : Community

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