Disclaimer: I am aware that this question is entirely moot, since any smooth manifold with boundary is homotopic to its interior! Nonetheless, I am still interested in learning what goes wrong if we don't use this fact.
The standard approach to deriving the Mayer-Vietoris sequence for de Rham cohomology uses an open cover $\{U,V\}$ of a smooth manifold $M$, together with the short exact sequence:
$$0\rightarrow \Omega^q(U \cup V) \xrightarrow{r^*} \Omega^q(U)\oplus\Omega^q(V) \xrightarrow{\iota_U^* - \iota_V^*} \Omega^q(U\cap V) \rightarrow 0$$
where $\iota_U: U\cap V \rightarrow U$ is the canonical inclusion (and similar for $V$), and the map $r^*$ concatenates the restrictions of forms from $M$ to $U$ and $V$. The non-trivial step is the verification of the surjectivity of $\iota_U^* - \iota_V^*$, which can be done by appealing to a partition of unity subordinate to $\{U,V\}$ to extend a differential form defined on the intersection $U\cap V$ to both $U$ and $V$.
I'm curious as to what goes wrong if we add boundaries to everything here. For simplicity assume compactness, so that the notions of topological and manifold boundary coincide. Let's suppose that $U$ and $V$ are still non-disjoint open sets, but they no longer cover our space $M$. Instead, suppose that $M$ is covered by the closures $\{\bar{U},\bar{V}\}$. Using the analogues of the maps mentioned above, we seem to be able to define the following sequence.
$$0\rightarrow \Omega^q(\bar{U} \cup \bar{V}) \xrightarrow{r^*} \Omega^q(\bar{U})\oplus\Omega^q(\bar{V}) \xrightarrow{\iota_{\bar{U}}^* - \iota_{\bar{V}}^*} \Omega^q(\bar{U}\cap \bar{V}) \rightarrow 0$$
I would think that something may go wrong when arguing for the exactness of this sequence. However, I will play devil's advocate and try to prove exactness directly:
- $r^*$ should still be injective, since any pair of global forms on $M$ have to differ on either $\bar{U}$ or $\bar{V}$.
- Any global form defined on $M$ will still be mapped to zero under the difference map, so $Im(r^*) \subseteq \ker{(\iota_{\bar{U}}^* - \iota_{\bar{V}}^*)}$. For the converse, if we have a pair of forms $\omega$ and $\eta$ that agree on the intersection $\bar{U}\cap \bar{V}$, it seems that we can just define a global form $\mu$ on $M$ to be the gluing of these two forms, i.e. $$ \mu(x) = \begin{cases} \omega(x) & \text{ if } x\in\bar{U} \\ \eta(x) & \text{ if } x\in\bar{V} \end{cases}$$ with $\mu$ being well-defined on the intersection due to the vanishing of the difference map, and smoothness coming from the smoothness of $\omega$ and $\eta$.
- No open sets for partitions of unity. This means that we cannot use the same argument to prove the surjectivity of the difference map. However, any form $\omega$ defined on the closed subset $\bar{U}\cap\bar{V}$ can be extended into $\bar{U}$ (or $\bar{V}$) using a partition of unity subordinate to the open set $\{A,B\}$, where $A$ is an outward-pointing collar neighbourhood of $\bar{U}\cap\bar{V}$ in $\bar{U}$, and $B$ is the open set $\bar{U} \backslash (\bar{U}\cap\bar{V})$. We can extend $\omega$ by zero, and then take the form $(\omega, 0)$ on $\Omega^q(\bar{U})\oplus\Omega^q(\bar{V})$. This seems to confirm the surjectivity of $\iota_{\bar{U}}^* - \iota_{\bar{V}}^*$
I guess that there is something wrong in the arguments above. Perhaps there might be some issue with the smoothness of the sets $\bar{U}$ and $\bar{V}$. By construction $U$ and $V$ are open, so $U\cap V$ is fine, however perhaps the boundary $\partial(U\cap V)$ is not smooth in general, so naively writing "$\Omega^q(\bar{U}\cap \bar{V})$" doesn't actually make sense.
Anyway, please let me know if you have any thoughts on this!