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Let $X$ be a CW complex. Suppose, $$\emptyset\subset X^0\subset X^1\subset \cdots \subset X^p\subset \cdots \subset X= \bigcup_{i=0}^{\infty} X^i,$$ is a filtration of $X$ by its skeletons $X^i$. Now how to get a module filtration of the cochain complex $C^*(X)$ out of this given skeleton filtration, so that we can get the cohomology spectral sequence $(E_r, d_r)$ of the module $C^*(X)$ associated with the (wanted) filtration such that this spectral sequence satisfies:

(i) First quadrant, (ii) $E_1^{p,q}\cong H^{p+q}(X^p,X^{p-1})$ and (iii) collapses at $E_2$.

I have tried defining the filtration of $C^*(X)$ as follows: Let $C_*(X)$ be the singular chain complex of $X$ filtered by $$0 \subset C_*(X^0)\subset C_*(X^1)\subset \cdots\subset C_*(X^p)\subset \cdots \subset C_*(X).$$ Now define the filtration $F^p$ of $C^*(X)$ by $$F^p = \operatorname{Ann}(C_*(X^{p-1})) = \{f\in C^*(X): f(C_*(X^{p-1}))=0\}.$$ The cohomology spectral sequence obtained from this filtration satisfies (i) above, but I cannot show whether it satisfies (ii) and (iii).

Is the above filtration of $C^*(X)$ correct? If yes how to prove part (ii) and (iii), and if it is not correct then how do we define the filtration of $C^*(X)$.

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    $\begingroup$ Take $F_pC_*(X) = C_*(X^p)$ $\endgroup$
    – Drew Heard
    Sep 30, 2021 at 15:01
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    $\begingroup$ For (ii) you should recall the definition of relative cohomology and for (iii) you should use the isomorphism $H^*(X,A) \cong \bar{H}^*(X/A)$. $\endgroup$ Sep 30, 2021 at 15:04
  • $\begingroup$ It is possible that a reference such a spectral sequence as is is difficult to find. However, it is actually a degenerated case of Atiyah-Hirzebruch spectral sequence, and as such, there are plenty of references. $\endgroup$
    – user43326
    Oct 4, 2021 at 8:42
  • $\begingroup$ @DrewHeard op is asking for a filtration in cohomology, not in homology $\endgroup$
    – user43326
    Oct 4, 2021 at 8:43

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