Explicit de rham mayer vietoris connecting class

Given: Assume countable choice. A closed overlap form $\omega$ and the partition lift $(\alpha ,\beta )=(-{\rho }_V\omega ,{\rho }_U\omega )$.

Mayer vietoris sequence in de rham cohomology: Under countable choice the de Rham Mayer–Vietoris sequence is exact: $\cdots \to H^k(M)\stackrel{r^{\ast }}{\to }{H}^k(U)\oplus H^k(V)\stackrel{s^{\ast }}{\to }{H}^k(U\cap V)\stackrel{\delta }{\to }{H}^{k+1}(M)\to \cdots$, beginning with $0\to H^0(M)$. Its proof obtains this $\delta$ by applying the long exact sequence theorem to the short exact de Rham cochain sequence in F3, with no additional differential sign.

The de rham mayer vietoris difference map is surjective: Assume countable choice. The difference map $s:{\mathrm{\Omega }}^k(U)\oplus {\mathrm{\Omega }}^k(V)\to {\mathrm{\Omega }}^k(U\cap V)$ is surjective in every degree.

Short exact mayer vietoris sequence of de rham complexes: Under countable choice, $0\to {\mathrm{\Omega }}^{\bullet }(M)\stackrel{r}{\to }{\mathrm{\Omega }}^{\bullet }(U)\oplus {\mathrm{\Omega }}^{\bullet }(V)\stackrel{s}{\to }{\mathrm{\Omega }}^{\bullet }(U\cap V)\to 0$ is short exact as a sequence of real cochain complexes, with $s(\alpha ,\beta )=\beta -\alpha$.

Elementwise formula for the connecting map in module categories: Let $R$ be a ring and let $0\to A_{\bullet }\stackrel{i}{\to }{B}_{\bullet }\stackrel{p}{\to }{C}_{\bullet }\to 0$ be a short exact sequence of chain complexes of left $R$-modules. If $[c]\in H_n(C)$ is represented by a cycle $c\in C_n$, choose a lift $b\in B_n$ with $p_n(b)=c$, and let $a\in A_{n-1}$ be the unique element satisfying $i_{n-1}(a)=d_n^B(b)\mathrm{.}$ Then ${\mathrm{\partial }}_n([c])=[a]\in H_{n-1}(A)\mathrm{.}$ This class is independent of the chosen lift $b$ and of the chosen cycle representative $c$.