Mayer–Vietoris sequence for de Rham cohomology

Statement

Let $M=U\cup V$ be a cover by two open sets of a smooth manifold. Restriction and difference of restrictions give a short exact sequence of de Rham complexes

0\longrightarrow\Omega^\bullet(M)\longrightarrow \Omega^\bullet(U)\oplus\Omega^\bullet(V)\longrightarrow \Omega^\bullet(U\cap V)\longrightarrow0.

The associated cohomology sequence is exact:

\cdots\to H_{\mathrm{dR}}^k(M)\to H_{\mathrm{dR}}^k(U)\oplus H_{\mathrm{dR}}^k(V) \to H_{\mathrm{dR}}^k(U\cap V)\xrightarrow{\delta} H_{\mathrm{dR}}^{k+1}(M)\to\cdots.

This is the Mayer–Vietoris sequence for de Rham cohomology. Its connecting homomorphism measures the obstruction to obtaining a class on $U\cap V$ as the difference of classes restricted from $U$ and $V$ .

Exactness and the connecting map

The last map of complexes sends $(\alpha,\beta)$ to $\alpha|_{U\cap V}-\beta|_{U\cap V}$ . Its surjectivity follows from a partition of unity subordinate to the cover. If a closed form $\omega$ on $U\cap V$ is the difference of restrictions of $(\alpha,\beta)$ , then $d\alpha$ and $d\beta$ agree on the overlap and glue to a global closed form $\eta$ . The connecting homomorphism is $\delta[\omega]=[\eta]$ .

Use in calculations

If $U$ , $V$ , and $U\cap V$ have known cohomology, exactness often determines the de Rham cohomology of $M$ . This is the differential-form analogue of the Mayer–Vietoris sequence in singular cohomology and is a principal gluing step in one proof of the de Rham theorem Bott and Tu, de Rham theory.

Naturality and sign convention

A smooth map compatible with two chosen covers induces a morphism of the corresponding long exact sequences. Reversing the difference map from $\alpha-\beta$ to $\beta-\alpha$ changes the displayed connecting map by a sign but does not change exactness. The convention must therefore be fixed when explicit representatives are compared.

References

Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: Chapter I, the Mayer–Vietoris sequence and de Rham theorem.
Loring W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: the chapter on de Rham theory and Mayer–Vietoris arguments.