Definition:De Rham Cohomology

Definition

Let $X$ be a smooth manifold.

Let $\struct {\Omega^\bullet, d}$ be the de Rham complex of $X$.

Then the de Rham Cohomology of $X$ is defined to be the cohomology groups of $\Omega^\bullet$.

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It is usually denoted by $\map {H^\bullet_{\text {de Rham} } } X$, $\map {H^\bullet_{\text {dR}} } X$ or when there is no ambiguity, $\map {H^\bullet} X$.

Also see

• Results about the de Rham cohomology can be found here.

Source of Name

This entry was named for Georges de Rham.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): de Rham cohomology
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): de Rham cohomology