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Suppose I have a topological space $X$ with a collection of closed subsets $X_\tau$ for $\tau \in P$ where I think of $P$ as a poset with $\tau \leq \lambda \iff X_\tau \subset X_\lambda$. Is there some nice (algorithmic?) way to get information about the (compactly supported) cohomology of $U_{\tau} = X_\tau - \bigcup_{\lambda<\tau}X_\lambda$ in terms of the cohomology of the $X_\tau$?

For instance, if we I just one closed subset $Z \subset X$, then there is a long exact sequence: $$\dots \to H^n_c(U) \to H^n_c(X) \to H^n_c(Z) \to \dots$$ Is there something similar in the general case? I can certainly think of some inductive process for building up the relevant cohomology starting from the lowest strata by a combination of the excision sequence and Mayer-Vietoris but it would be nice if there was succinct description.

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  • $\begingroup$ Is $P$ the collection of all closed subsets or just some poset? To recover cohomology you need much more than just the collection of cohomologies, but also induced homomorphisms. $\endgroup$ Jul 29 at 23:04
  • $\begingroup$ $P$ is just some poset (that I know a lot about). Let's say I know the induced homomorphisms too. $\endgroup$
    – Asvin
    Jul 29 at 23:08
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    $\begingroup$ I think you can obtain a spectral sequence where the first term is the cohomology of $X_\tau$, the second term is the sum over all (let's say) maximal $\lambda < \tau$ of the cohomology of $X_\lambda$, the third is the sum over all pairs of maximal $\lambda_1,\lambda_2<\tau$ of the cohomology of $X_{\lambda_1} \cap X_{\lambda_2}$, and so on. (All these cohomologies are compactly supported). Whether that's computable depends a lot on your spaces and maps... $\endgroup$
    – Will Sawin
    Jul 30 at 0:28
  • $\begingroup$ @WillSawin That's along the lines of what I am imagining too. Do you have any references in mind where this might have been worked out or an example along these lines might have occurred? $\endgroup$
    – Asvin
    Jul 30 at 1:02
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    $\begingroup$ Your example may be a special case of the theory of hypercoverings as in Deligne's construction of mixed Hodge theory (or other works you can find by searching around, e.g. Brian Conrad's notes on cohomological descent math.stanford.edu/~conrad/papers/hypercover.pdf). The proof I would thin of would be to write down an explicit long exact sequence of sheaves starting with the constant sheaf on $X_\tau$, then the sum of the constant sheaves on $X_\lambda$, etc., then view it as a filtration of an object in the derived category of sheaves and take the spectral sequence of a filtration. $\endgroup$
    – Will Sawin
    Jul 30 at 1:10
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I wrote a paper on precisely this question:

A spectral sequence for stratified spaces and configuration spaces of points. Geom. Topol. 21 (2017), no. 4, 2527–2555.

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  • $\begingroup$ Perfect, thanks! $\endgroup$
    – Asvin
    Jul 30 at 18:28
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I am still unsure about your assumptions on $P$ and on the amount of information that you have. Here is a silly example:

Let $X=X_\tau$ be, say, ${\mathbb R}^n$ and take the posets $P=\{X, \{x\}\}$, $Q=\{X, \{x\}, \{y\}\}$, where $x\ne y$ are points in $X$. Then, of course, $H^*_c(X\setminus \{x,y\})$ is not isomorphic to $H^*_c(X\setminus \{x\})$, even though you have the same data in terms of induced homomorphisms $$ H^*_c(X)\to H^*_c(X_\nu), $$ for $X_\nu \in P$ and for $X_\nu \in Q$.

As Will suggested, if you have much more data in terms of the boolean algebra generated by $P$ and the corresponding induced homomorphisms, then you can build a spectral sequence.

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  • $\begingroup$ In my case, the $X_\tau$ will just be projective spaces of varying dimensions. I also suspect that there should be spectral sequence formulation. Do you have a reference for where I can read about this or even where an example is done of this kind so I don't have to reinvent the wheel? $\endgroup$
    – Asvin
    Jul 30 at 1:01
  • $\begingroup$ If I understand your example right, doesn't it fall under the excision exact sequence category? I am happy getting partial information about my cohomology groups if that's what you mean. $\endgroup$
    – Asvin
    Jul 30 at 1:04

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