The Mayer–Vietoris Sequence

As the example of H1(R) illustrates, calculating the de Rham cohomology of a manifold amounts to solving a canonically given system of differential equations on the manifold and in case it is not solvable, to finding the obstructions to its solvability. This is usually quite difficult to do directly. We introduce in this chapter one of the most useful tools in the calculation of de Rham cohomology, the Mayer-Vietoris sequence. Another tool, the homotopy axiom, will come in the next chapter.