Advertisement

The Čech-de Rham Complex

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

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 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].

Keywords

Vector Bundle Open Cover Cohomology Class Good Cover Euler Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations