The Čech-de Rham Complex

Abstract

Let U and V be open sets on a manifold. In Section 2, we saw that the sequence of inclusions $$U \cup V \leftarrow U\coprod V \Leftarrow U \cap V$$ gives rise to an exact sequence of differential complexes $$0 \to \Omega *(U \cup V) \to \Omega *(U) \oplus \Omega *(V) \to \Omega *(U \cap V) \to 0$$ called the Mayer—Vietoris sequence. The associated long exact sequence $$\cdot \cdot \cdot \to {H^q}(U \cup V){H^q}(U) \oplus {H^q}(V){H^q}(U \cap V){H^{q + 1}}(U \cup V) \to \cdot \cdot \cdot $$ allows one to compute in many cases the cohomology of the union U ∪ V from the cohomology of the open subsets U and V. In this section, the Mayer-Vietoris sequence will be generalized from two open sets to countably many open sets. The main ideas here are due to Weil [1].