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For compact support cohomology, the Mayer-Vietoris sequence is quite different. First of all, there is no functoriality, that is, maps between manifolds do not induce maps over forms. But there is hope for inclussions - and in a way that has no non-compact equivalent

Let $M$ be a manifold and $U$ an open set of $M$. The inclussion induces the map $i_*:\Omega^k_c(U)\longrightarrow\Omega^k_c(M)$ (note the opposite direction!), defined by

$$i_*(\alpha)= \begin{cases} \alpha & \text{in \(U\)}\\ 0 & \text{in \(M-\text{sop}(\alpha)\)} \end{cases}.$$

That is, since $\alpha$ has compact support, its support doesn't reach the "border" of $U$ and can be extended by zero differentiably

And again we may gather everything up and build this chain: