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 $\Omega^\bullet$.

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