Skip to main content

The Čech-de Rham Complex

  • Chapter

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.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-1-4757-3951-0_3
  • Chapter length: 65 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   59.99
Price excludes VAT (USA)
  • ISBN: 978-1-4757-3951-0
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   74.99
Price excludes VAT (USA)
Hardcover Book
USD   74.95
Price excludes VAT (USA)

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 1982 Springer Science+Business Media New York

About this chapter

Cite this chapter

Bott, R., Tu, L.W. (1982). The Čech-de Rham Complex. In: Differential Forms in Algebraic Topology. Graduate Texts in Mathematics, vol 82. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3951-0_3

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-3951-0_3

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2815-3

  • Online ISBN: 978-1-4757-3951-0

  • eBook Packages: Springer Book Archive