Better spelling "DeRham", not derham... I can't figure out how to change this... moderators? The cohomology of the complex of differential forms on a smooth manifold with differential given by exterior derivative.
Given a smooth $n$-dimensional manifold $M$, there is a complex
$$0 \to \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \Omega^2(M) \xrightarrow{d} \dots \xrightarrow{d} \Omega^{n-1}(M) \xrightarrow{d} \Omega^n(M) \to 0$$
called the de Rham complex. The cohomology of this complex is called de Rham cohomology: $H^k_{\text{dR}}(M) = H^k(\Omega^{\bullet}(M), d)$.
One part of the Hodge Theorem states that if $M$ is compact, $H^k_{\text{dR}}(M)$ is a finite-dimensional vector space for every $k$.
Wikipedia article: De Rahm cohomology
