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.



Can I reconstruct the cohomology from a collection of open sets?
I wrote a paper on precisely this question:


[A spectral sequence for stratified spaces and configuration spaces of points.](https://msp.org/gt/2017/21-4/p16.xhtml)
Geom. Topol. 21 (2017), no. 4, 2527–2555.


