---- > [!theorem] Theorem. ([[de Rham's theorem]]) > The [[de Rham cohomology]] and [[singular cohomology]] of a [[smooth manifold|smooth]] $n$-[[smooth manifold|manifold]] $M$ agree, per the isomorphism $\begin{align} H^{n}_{\text{dR}}(M) & \xrightarrow{\sim} H^{n}(M) \\ [\omega] & \mapsto \left[ \sigma \mapsto \int _{\sigma} \omega \right] \end{align}$ ^theorem > [!proof]- Proof. ([[de Rham's theorem]]) > The proof here uses [[sheaf cohomology]], which seems fun: https://tlovering.wordpress.com/wp-content/uploads/2011/04/sheaftheory.pdf ---- #### ----- #### 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 > ```