---- > [!definition] Definition. ([[de Rham cohomology]]) > Let $M$ be a [[smooth manifold]] of dimension $n$. For $r \in \mathbb{Z}_{\geq 0}$ denote by $\Omega^{r}(M)$ the space of [[differential form|differential]] $r$-[[differential form|forms]] on $M$. > The property $d_{r} \circ d_{r-1} = 0$ of [[exterior derivative]] yields a [[chain complex of modules|cochain complex]], called the **de Rham cochain complex**: $0 \to \Omega^{0}(M) \xrightarrow{d_{0}}\Omega^{1}(M) \xrightarrow{d_{1}} \Omega^{2}(M) \xrightarrow{d_{2}} \cdots$ the [[chain complex of modules|cohomology]] of which is called the **de Rham cohomology of $M$** and denoted $H^{\bullet}_{\text{dR}}(M)$. ^definition > [!basicexample] > If $M$ is [[connected]], then $H_{\text{dR}}^{0}(M)\cong \mathbb{R}$. Indeed, functions $f \in C^{\infty}(M)$ with $df=0$ are precisely the locally constant (=constant because $M$ is connected) functions. There is one constant function $M \to \mathbb{R}$ per element of $\mathbb{R}$. Hence the identification. > > If $M$ is [[connected]], [[compact]], and [[orientable manifold|oriented]] then $H_{\text{dR}}^{\text{dim } M} \cong \mathbb{R}$. Indeed, the isomorphism is given by $\omega \mapsto \int _{M} \omega$. Certainly such a map is surjective. Its [[kernel]] is those forms which [[integration of a compactly supported volume form on an oriented smooth manifold|integrate]] to zero, which we know is precisely the exact forms. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```