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

Question

Let $F \to E \to B$ be a fibration with $B$ simply connected (more generally, such that $\pi_1(B)$ acts trivially on the homology of $F$). Then there is a Serre spectral sequence $H_p(B, H_q(F)) \to H_{p+q}(E)$. 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 $\mathbb{Z}$.

But there is another spectral sequence for sheaf cohomology: the Leray spectral sequence. Given spaces $X, Y$ and $f: X \to Y$, and a sheaf $\mathcal{F}$ on $X$, there is a spectral sequence $H^p(Y, R^q_f(\mathcal{F})) \to H^{p+q}(X, \mathcal{F})$. 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?

Answer 1

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 $\Omega X \rightarrow PX \rightarrow X$, where $\Omega X$ is the loop space of $X$ and $PX$ is the space of paths starting at the basepoint of $X$ and the map $PX \rightarrow X$ 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!

Answer 2

This is not intended to be an answer but rather a long comment.

First, on the question why does Serre take all the credit if his theorem is a particular case of Leray's?. Well, John McCleary says on page 139 of his book User's Guide to Spectral Sequences (2ed):

For the Cêch or Alexander-Spanier cohomology theories, the multiplicative structure is carried along transparently in the construction of the spectral sequence and so we get a spectral sequence of algebras directly with converges to $H^*(E;R)$ as an algebra. The result for singular theory, however, is more difficult --- it is one of the technical triumphs of Serre's celebrated thesis.

And, as the same wiki page that the OP cites say:

Earlier (1948/9) the implications for fiber bundles were extracted in a form formally identical to that of the Serre spectral sequence, which makes no use of sheaves. This treatment, however, applied to Alexander–Spanier cohomology with compact supports (....). Jean-Pierre Serre, who needed a spectral sequence in homology that applied to path space fibrations, whose total spaces are almost never locally compact, thus was unable to use the original Leray spectral sequence (...).

On another topic, and as @T_P points out, Serre's original result does not impose any condition on the homotopy group of the base. That assumption is added so that one can simply the $E_2$-term by getting rid of (co)homology with local coefficients.

Finally, I would like to say a few words about spectral sequences and their initial terms. Apparently, there is a Grotehndieck's spectral sequence whose $E_2$-term is exactly as in Serre's, but the equality of both is questioned, since nothing is said about the differentials. This is completely true. However, there are (not a few) case where sheaf cohomology and singular cohomology provide the same result. Thus, one is found dealing with two spectral sequences which have the same initial term and the same limit. This is still not enough to guarantee that both are the same spectral sequence in general. However, for the particular case when one of them degenerates, so must the other; in this case both become eventually the same. I know it is a very particular situation, but it is also one that becomes very handy sometimes.