# Topological Derivation of Leray Spectral Sequence

I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is almost, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step currently is writing down the edge maps on the $E_2$ page.

Can someone point me to a reference that derives the Leray sequence from a topological/geometric point of view (i.e. not by appealing to the Grothendieck spectral sequence)? I recall reading somewhere that a CW-structure on the base space can induce the needed filtration, but the reference I found (which I can't remember now) only did this for the case of an actual fiber bundle.

• Why don't you want to use the Grothendieck spectral sequence? Intuitively it should generalize much more easily than any topological method. The topological method is usually used only for the special case of fiber bundles and local coefficients (that is locally constant sheaves) and I don't know how much of it can be generalized. – Denis Nardin Feb 11 '16 at 23:21
• Indeed, the Grothendieck spectral sequence tells you that you need to understand the derived pushforward. In good cases (presumably including the one you're talking about) the derived pushforward of the constant sheaf isn't quite a local system but you can stratify the base so that it's a local system on each stratum (I.e. You have a constructible sheaf) and then you can try to compute from there. – Dylan Wilson Feb 12 '16 at 3:45
• Thanks! In my case, I understand the derived pushforward reasonably well. The thing I don't understand is the edge map itself, especially on the torsion subgroup. I know the form of the sequence, but the Grothendieck formalism makes it hard to see what the ($E_2$) maps are. I want something like the CW-complex answer for fibrations. (If it helps: in my case, all fibers are $S^1$, but they have multiplicity at some points) – dorebell Feb 12 '16 at 4:00
• I'm rather ignorant about 3-manifolds, but I thought a Seifert fiber space was an honest-to-God fiber bundle (tho' over an orbifold). And the sheaf-theoretic derivation/interpretation works just as well for orbifolds/stacks. – Dan Petersen Feb 12 '16 at 6:32
• Any Serre fibration over a finite CW base is fiberwise weak equivalent to a fiber bundle. So I don't understand what the issue is.... – John Klein Feb 12 '16 at 11:17

## 2 Answers

Bott and Tu do this in their book Differential forms in algebraic topology, see Section 14, Leray's construction" (starting on page 179).

• Wonderful: exactly what I was looking for. I didn't realize they did the general case there. – dorebell Feb 16 '16 at 7:15

One such reference is

Segal's "Classifying Spaces and Spectral Sequences"

See proposition 5.2.