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Just as we did with the sphere $\mathbb{S}^2$, we would like to compute the cohomology of a manifold $M$ union of smaller manifolds, say two open sets $U$, $V$. The inclusions among the different spaces induce reversed arrows for forms: one just has to restrict the forms to their new domain

And we may gather everything up and build this chain:

But this chain happens to be very useful