Naturality of de rham mayer vietoris for maps of covered manifolds

Given: Assume countable choice. Open covers $M=U\cup V$, $N=U\mathrm{\prime }\cup V\mathrm{\prime }$ and the stated smooth covered map $F$.

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 $0\to H^0(M)$.

Explicit de rham mayer vietoris connecting class: Under countable choice, for a closed $k$-form $\omega$ on $U\cap V$, $\delta [\omega ]=[\eta ]$, where $\eta {\mathrm{\mid }}_U=d(-{\rho }_V\omega )$ and $\eta {\mathrm{\mid }}_V=d({\rho }_U\omega )$, with products smoothly extended by zero as in the lift construction. This class is independent of partition, lift and representative.

Pullback of forms is smooth functorial and preserves wedges: For a smooth map $F:M\to N$, pullback sends smooth differential forms on $N$ to smooth differential forms on $M$, is functorial, and satisfies $F^{\ast }(\alpha \wedge \beta )=F^{\ast }\alpha \wedge F^{\ast }\beta \mathrm{.}$

Naturality of the homology connecting morphism: A morphism of short exact sequences of complexes induces a commutative square $\begin{array}{ccc}{H}_n(C)& \stackrel{{\mathrm{\partial }}_n}{\to }& H_{n-1}(A)\\ \downarrow & & \downarrow \\ H_n(C^{\mathrm{\prime }})& \stackrel{{\mathrm{\partial }}_n^{\mathrm{\prime }}}{\to }& H_{n-1}(A^{\mathrm{\prime }})\end{array}$ for every $n\in \mathbb{Z}$.

Pullback is a morphism of de rham complexes: A smooth map $F:M\to N$ induces a degree-zero real cochain map $F^{\ast }:{\mathrm{\Omega }}^{\bullet }(N)\to {\mathrm{\Omega }}^{\bullet }(M)$.