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Leray's theory of spectral sequences considers a continuous map $f : X \to Y$ between topological spaces. The statement is that there exists a filtration $$H^n(X,k) = F^0H^n(X,k) \supseteq ... \supseteq F^nH^n(X,k),$$ $k$ being a ring, such that the subquotients of this filtration equals the terms $E_{\infty}^{p,n-p}$ of a spectral sequence which has $E_2$-term $$E_2^{p,q} = H^p(Y,R^qf_* k),$$ where $R^qf_*k$ denotes the sheafification of the presheaf $U \to H^q(f^{-1}(U),k)$.

Where can I find a genuine construction of this filtration ? Is there a way to express it in terms of the skeletons of $Y$ (in the CW-complex case), as for the Serre spectral sequence ? I would like to check that a certain homomorphism of cochain complexes induces a map between Leray spectral sequences, and thus I need to check that the map in question preserves this filtration.

Thanks in advance.

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  • $\begingroup$ Bott and Tu's book is quite explicit with the Leray spectral sequence for local systems on top. spaces. $\endgroup$ – user40276 Aug 31 '18 at 20:54
  • $\begingroup$ Bott and Tu’s book defines a double complex based on the Cech complex and the complex of differential forms. In particular it supposes that the spaces are smooth manifolds, which I don’t want to assume. $\endgroup$ – BrianT Aug 31 '18 at 21:22
  • $\begingroup$ Actually, they consider also fibrations over topological spaces and locally constant sheaves. See the next chapters. I don't have access to the book now, so I can't check the exact page. $\endgroup$ – user40276 Sep 1 '18 at 21:09
  • $\begingroup$ I have checked again, and in any case they don’t treat the general case of continuous maps which are not fibration. $\endgroup$ – BrianT Sep 2 '18 at 7:38
  • $\begingroup$ The general case works in the same way except that the second page won't be the cohomology of the constant sheaf $H^q (F, k)$. I may write the details later. Right now, I'm out of time. $\endgroup$ – user40276 Sep 2 '18 at 11:29

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