The Čech-de Rham Complex

  • Raoul Bott
  • Loring W. Tu
Part of the Graduate Texts in Mathematics book series (GTM, volume 82)


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 MayerVietoris 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 UV 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].


Vector Bundle Open Cover Cohomology Class Good Cover Euler Class 
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Copyright information

© Springer Science+Business Media New York 1982

Authors and Affiliations

  • Raoul Bott
    • 1
  • Loring W. Tu
    • 2
  1. 1.Mathematics DepartmentHarvard UniversityCambridgeUSA
  2. 2.Department of MathematicsTufts UniversityMedfordUSA

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