[Verse] In mathematics The de Rham cohomology Or de Rham cohomology Is a tool used in algebraic topology To study the topology Of smooth manifolds Using differential forms [Chorus] It is named after Georges de Rham The cohomology groups are denoted By H with a superscript of k And a subscript of dR H dR k and the k-th cohomology group And for k greater than or equal to zero [Verse] A result that goes back to Hermann Weyl States that the de Rham cohomology Of a closed manifold Is isomorphic to the Čech cohomology Of its sheaf of locally constant functions With values in R [Verse] The integration of differential forms On chains establishes a duality Between de Rham cohomology And the homology of chains The Poincaré lemma implies that On a contractible smooth manifold The de Rham cohomology is Isomorphic to R Concentrated in degree zero [Chorus] This isomorphism is also referred to As a quasi-isomorphism If M is a smooth manifold of dimension n And Omega M is the differential graded algebra Of differential forms on M The exterior derivative d on differential forms Is a differential that increases degree by one And satisfies d squared equals zero So it makes Omega M A differential complex And the k-th de Rham cohomology H dR k is the k-th cohomology of this complex It is an R vector space [Verse] The degree zero cohomology group H dR zero is the vector space of smooth functions On M that are locally constant That is to say the kernel of d from Zero forms to one forms In the de Rham complex This vector space Is canonically isomorphic to The vector space of locally constant functions And the integral cohomology H of M By the Poincaré lemma The isomorphism is given By integrating over M The de Rham cohomology class Of a closed differential form That represents a class in the de Rham cohomology
