I am interested whether there exists a versions of de Rham relative cohomology $H^\bullet(M, N)$ in which $N$ does not need to be a manifold. I know there are two main definitions in literature as described in this question: Relative De Rham cohomologies. The obvious thing would be to require $\omega \in \Omega^\bullet(M)$ to vanish not only in the direction of $N$ but in some open neighborhood of $N$. This sounds to me like a sheaf-based construction (and also remains me a little bit of Alexander Spanier cohomology) but I couldn't find it anywhere. Can this be found in the literature?
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$\begingroup$ Why would you require the form to vanish on a whole neighborhood of $N$? That's not asked even when $N$ is a manifold. For example if $M = \mathbb{R}$ and $N = \{0\}$ (a submanifold), then $\Omega^0(M,N)$ are functions on $\mathbb{R}$ that vanish at $0$, not necessarily in a neighborhood of $0$. $\endgroup$– Najib IdrissiCommented Aug 5 at 0:46
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$\begingroup$ @NajibIdrissi At least, for compact submanifolds, it does not seem to hurt by the tubular neighborhood theorem, I think. $\endgroup$– Z. MCommented Aug 5 at 6:12
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1$\begingroup$ Two model examples to test your definitions (whether they give an answer which agrees with Spanier or simplicial cohomology) would be plane without a converging sequence and plane without a Cantor set. If you look at the complex formed by germs of forms supported away from a closed subset, you will (probably) get something closer to the 'right' answer them in the case of forms actually supported away from a closed subset. $\endgroup$– Denis TCommented Aug 5 at 8:39
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