# Differentiable Manifolds/De Rham cohomology

Proposition (the differentiable forms of a differentiable manifold and the Cartan derivative constitute a cochain complex):

Let $M$ be a differentiable manifold of class ${\mathcal {C}}^{k}$ . Then the diagram

$0\longrightarrow \Omega ^{0}(M){\overset {d}{\longrightarrow }}\Omega ^{1}(M){\overset {d}{\longrightarrow }}\Omega ^{2}(M){\overset {d}{\longrightarrow }}\cdots$ constitutes a chain complex of modules over ${\mathcal {C}}^{k}(M)$ , where $d$ shall denote the Cartan derivative.

Proof: This follows immediately from the fact that applying the Cartan derivative twice always yields zero. $\Box$ Definition (de Rham cohomology):

Let $M$ be a differentiable manifold of class ${\mathcal {C}}^{k}$ . The cohomology arising from the chain complex

$0\longrightarrow \Omega ^{0}(M){\overset {d}{\longrightarrow }}\Omega ^{1}(M){\overset {d}{\longrightarrow }}\Omega ^{2}(M){\overset {d}{\longrightarrow }}\cdots$ is called de Rham cohomology. The $k$ -th ${\mathcal {C}}^{k}(M)$ -module of this cohomology is commonly denoted $H_{\text{dR}}^{k}(M)$ .