de Rham Cohomology

de Rham Complex

de Rham Complex

Let $M$ be a smooth manifold. The de Rham complex is the sequence of differential forms on $M$: $0\to {\Omega }^0(M)\stackrel{\text{d}}{\to }{\Omega }^1(M)\stackrel{\text{d}}{\to }{\Omega }^2(M)\stackrel{\text{d}}{\to }\cdots$where ${\Omega }^k(M)$ is the space of smooth differential $k$-forms on $M$, and $\text{d}$ is the exterior derivative.

de Rham Cohomology Group

The $k$th de Rham cohomology group of a smooth manifold $M$ is defined as the quotient $H_{\mathrm{dR}}^k(M)=\frac{\ker (\text{d}:{\Omega }^k(M)\to {\Omega }^{k+1}(M))}{\mathrm{im}(\text{d}:{\Omega }^{k-1}(M)\to {\Omega }^k(M))}.$ That is, it’s the space of closed $k$-forms modulo the space of exact $k$-forms.

de Rham’s theorem

de Rham's Theorem

Let $M$ be a smooth manifold. Then the de Rham cohomology groups $H_{\mathrm{dR}}^k(M)$ are isomorphic to the singular cohomology groups $H^k(M,\mathbb{R})$ with real coefficients. In particular, they are independent of the choice of differential forms used to define them.