Inmathematics,de Rham cohomology (named afterGeorges de Rham) is a tool belonging both toalgebraic topology and todifferential topology, capable of expressing basic topological information aboutsmooth manifolds in a form particularly adapted to computation and the concrete representation ofcohomology classes. It is acohomology theory based on the existence ofdifferential forms with prescribed properties.

On any smooth manifold, everyexact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of"holes" in the manifold, and thede Rham cohomology groups comprise a set oftopological invariants of smooth manifolds that precisely quantify this relationship.^[1]

The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples ofcohomology, namelyde Rham cohomology, which (roughly speaking) measures precisely the extent to which thefundamental theorem of calculus fails in higher dimensions and on general manifolds.
— Terence Tao, Differential Forms and Integration^[2]

Thede Rham complex is thecochain complex ofdifferential forms on somesmooth manifoldM, with theexterior derivative as the differential:

${\displaystyle 0\to {\mathrm{\Omega }}^0(M)\stackrel{d}{\to }{\mathrm{\Omega }}^1(M)\stackrel{d}{\to }{\mathrm{\Omega }}^2(M)\stackrel{d}{\to }{\mathrm{\Omega }}^3(M)\to \cdots ,}$

whereΩ⁰(M) is the space ofsmooth functions onM,Ω¹(M) is the space of1-forms, and so forth. Forms that are the image of other forms under theexterior derivative, plus the constant0 function inΩ⁰(M), are calledexact and forms whose exterior derivative is0 are calledclosed (seeClosed and exact differential forms); the relationshipd² = 0 then says that exact forms are closed.

In contrast, closed forms are not necessarily exact. An illustrative case is a circle as a manifold, and the1-form corresponding to the derivative of angle from a reference point at its centre, typically written asdθ (described atClosed and exact differential forms). There is no functionθ defined on the whole circle such thatdθ is its derivative; the increase of2π in going once around the circle in the positive direction implies amultivalued functionθ. Removing one point of the circle obviates this, at the same time changing the topology of the manifold.

One prominent example when all closed forms are exact is when the underlying space iscontractible to a point or, more generally, if it issimply connected (no-holes condition). In this case the exterior derivative${\displaystyle d}$ restricted to closed forms has a local inverse called ahomotopy operator.^[3][4] Since it is alsonilpotent,^[3] it forms a dualchain complex with the arrows reversed^[5] compared to the de Rham complex. This is the situation described in thePoincaré lemma.

The idea behind de Rham cohomology is to defineequivalence classes of closed forms on a manifold. One classifies two closed formsα,β ∈ Ω^k(M) ascohomologous if they differ by an exact form, that is, ifα −β is exact. This classification induces an equivalence relation on the space of closed forms inΩ^k(M). One then defines thek-thde Rham cohomology group${\displaystyle H_{\mathrm{d}\mathrm{R}}^k(M)}$ to be the set of equivalence classes, that is, the set of closed forms inΩ^k(M) modulo the exact forms.

Note that, for any manifoldM composed ofm disconnected components, each of which isconnected, we have that

${\displaystyle H_{\mathrm{d}\mathrm{R}}^0(M)\cong {\mathbb{R}}^m.}$

This follows from the fact that any smooth function onM with zero derivative everywhere is separately constant on each of the connected components ofM.

One may often find the general de Rham cohomologies of a manifold using the above fact about the zero cohomology and aMayer–Vietoris sequence. Another useful fact is that the de Rham cohomology is ahomotopy invariant. While the computation is not given, the following are the computed de Rham cohomologies for some commontopological objects:

For then-sphere,${\displaystyle S^n}$, and also when taken together with a product of open intervals, we have the following. Letn > 0,m ≥ 0, andI be an open real interval. Then

The${\displaystyle n}$-torus is the Cartesian product:${\displaystyle T^n=\underset{n}{\underbrace{S^1\times \cdots \times S^1}}}$. Similarly, allowing${\displaystyle n\ge 1}$ here, we obtain

${\displaystyle H_{\mathrm{d}\mathrm{R}}^k(T^n)\simeq {\mathbb{R}}^{(\genfrac{}{}{0ex}{}{n}{k})}.}$

We can also find explicit generators for the de Rham cohomology of the torus directly using differential forms. Given a quotient manifold${\displaystyle \pi :X\to X/G}$ and a differential form${\displaystyle \omega \in {\mathrm{\Omega }}^k(X)}$ we can say that${\displaystyle \omega }$ is${\displaystyle G}$-invariant if given any diffeomorphism induced by${\displaystyle G}$,${\displaystyle \cdot g:X\to X}$ we have${\displaystyle (\cdot g{)}^{\ast }(\omega )=\omega }$. In particular, the pullback of any form on${\displaystyle X/G}$ is${\displaystyle G}$-invariant. Also, the pullback is an injective morphism. In our case of${\displaystyle {\mathbb{R}}^n/{\mathbb{Z}}^n}$ the differential forms${\displaystyle d{x}_i}$ are${\displaystyle {\mathbb{Z}}^n}$-invariant since${\displaystyle d(x_i+k)=d{x}_i}$. But, notice that${\displaystyle x_i+\alpha }$ for${\displaystyle \alpha \in \mathbb{R}}$ is not an invariant${\displaystyle 0}$-form. This with injectivity implies that

${\displaystyle [d{x}_i]\in H_{dR}^1(T^n)}$

Since the cohomology ring of a torus is generated by${\displaystyle H^1}$, taking the exterior products of these forms gives all of the explicitrepresentatives for the de Rham cohomology of a torus.

Punctured Euclidean space is simply${\displaystyle {\mathbb{R}}^n}$ with the origin removed.

We may deduce from the fact that theMöbius strip,M, can bedeformation retracted to the1-sphere (i.e. the real unit circle), that:

${\displaystyle H_{\mathrm{d}\mathrm{R}}^k(M)\simeq H_{\mathrm{d}\mathrm{R}}^k(S^1).}$

Stokes' theorem is an expression ofduality between de Rham cohomology and thehomology ofchains. It says that the pairing of differential forms and chains, via integration, gives ahomomorphism from de Rham cohomology${\displaystyle H_{\mathrm{d}\mathrm{R}}^k(M)}$ tosingular cohomology groups${\displaystyle H^k(M;\mathbb{R}).}$ de Rham's theorem, proved byGeorges de Rham in 1931, states that for a smooth manifoldM, this map is in fact anisomorphism.

More precisely, consider the map

${\displaystyle I:H_{\mathrm{d}\mathrm{R}}^p(M)\to H^p(M;\mathbb{R}),}$

defined as follows: for any${\displaystyle [\omega ]\in H_{\mathrm{d}\mathrm{R}}^p(M)}$, letI(ω) be the element of${\displaystyle \text{Hom}(H_p(M),\mathbb{R})\simeq H^p(M;\mathbb{R})}$ that acts as follows:

${\displaystyle H_p(M)\ni [c]⟼{\int }_c\omega .}$

The theorem of de Rham asserts that this is an isomorphism between de Rham cohomology and singular cohomology.

Theexterior product endows thedirect sum of these groups with aring structure. A further result of the theorem is that the twocohomology rings are isomorphic (asgraded rings), where the analogous product on singular cohomology is thecup product.

For any smooth manifoldM, let$\underset{\_}{\mathbb{R}}$ be theconstant sheaf onM associated to the abelian group$\mathbb{R}$; in other words,$\underset{\_}{\mathbb{R}}$ is the sheaf of locally constant real-valued functions onM. Then we have anatural isomorphism

${\displaystyle H_{\mathrm{d}\mathrm{R}}^{\ast }(M)\cong H^{\ast }(M,\underset{\_}{\mathbb{R}})}$

between the de Rham cohomology and thesheaf cohomology of$\underset{\_}{\mathbb{R}}$. (Note that this shows that de Rham cohomology may also be computed in terms ofČech cohomology; indeed, since every smooth manifold isparacompactHausdorff we have that sheaf cohomology is isomorphic to the Čech cohomology${\check{H}}^{\ast }(\mathcal{U},\underset{\_}{\mathbb{R}})$ for anygood cover$\mathcal{U}$ ofM.)

The standard proof proceeds by showing that the de Rham complex, when viewed as a complex of sheaves, is anacyclic resolution of$\underset{\_}{\mathbb{R}}$. In more detail, letm be the dimension ofM and let${\mathrm{\Omega }}^k$ denote thesheaf of germs of${\displaystyle k}$-forms onM (with${\mathrm{\Omega }}^0$ the sheaf of$C^{\mathrm{\infty }}$ functions onM). By thePoincaré lemma, the following sequence of sheaves is exact (in theabelian category of sheaves):

${\displaystyle 0\to \underset{\_}{\mathbb{R}}\to {\mathrm{\Omega }}^0\stackrel{d_0}{\to }{\mathrm{\Omega }}^1\stackrel{d_1}{\to }{\mathrm{\Omega }}^2\stackrel{d_2}{\to }\cdots \stackrel{d_{m-1}}{\to }{\mathrm{\Omega }}^m\to 0.}$

Thislong exact sequence now breaks up intoshort exact sequences of sheaves

${\displaystyle 0\to \mathrm{i}\mathrm{m}{d}_{k-1}\stackrel{\subset }{\to }{\mathrm{\Omega }}^k\stackrel{d_k}{\to }\mathrm{i}\mathrm{m}{d}_k\to 0,}$

where by exactness we have isomorphisms$\mathrm{i}\mathrm{m}{d}_{k-1}\cong \mathrm{k}\mathrm{e}\mathrm{r}{d}_k$ for allk. Each of these induces a long exact sequence in cohomology. Since the sheaf${\mathrm{\Omega }}^0$ of$C^{\mathrm{\infty }}$ functions onM admitspartitions of unity, any${\mathrm{\Omega }}^0$-module is afine sheaf; in particular, the sheaves${\mathrm{\Omega }}^k$ are all fine. Therefore, the sheaf cohomology groups$H^i(M,{\mathrm{\Omega }}^k)$ vanish for$i>0$ since all fine sheaves on paracompact spaces are acyclic. So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms. At one end of the chain is the sheaf cohomology of$\underset{\_}{\mathbb{R}}$ and at the other lies the de Rham cohomology.

The de Rham cohomology has inspired many mathematical ideas, includingDolbeault cohomology,Hodge theory, and theAtiyah–Singer index theorem. However, even in more classical contexts, the theorem has inspired a number of developments. Firstly, theHodge theory proves that there is an isomorphism between the cohomology consisting of harmonic forms and the de Rham cohomology consisting of closed forms modulo exact forms. This relies on an appropriate definition of harmonic forms and of the Hodge theorem. For further details seeHodge theory.

IfM is acompactRiemannian manifold, then each equivalence class in${\displaystyle H_{\mathrm{d}\mathrm{R}}^k(M)}$ contains exactly oneharmonic form. That is, every member${\displaystyle \omega }$ of a given equivalence class of closed forms can be written as

${\displaystyle \omega =\alpha +\gamma }$

where${\displaystyle \alpha }$ is exact and${\displaystyle \gamma }$ is harmonic:${\displaystyle \mathrm{\Delta }\gamma =0}$.

Anyharmonic function on a compact connected Riemannian manifold is a constant. Thus, this particular representative element can be understood to be an extremum (a minimum) of all cohomologously equivalent forms on the manifold. For example, on a2-torus, one may envision a constant1-form as one where all of the "hair" is combed neatly in the same direction (and all of the "hair" having the same length). In this case, there are two cohomologically distinct combings; all of the others are linear combinations. In particular, this implies that the 1stBetti number of a2-torus is two. More generally, on an${\displaystyle n}$-dimensional torus${\displaystyle T^n}$, one can consider the various combings of${\displaystyle k}$-forms on the torus. There are${\displaystyle n}$ choose${\displaystyle k}$ such combings that can be used to form the basis vectors for${\displaystyle H_{\text{dR}}^k(T^n)}$; the${\displaystyle k}$-th Betti number for the de Rham cohomology group for the${\displaystyle n}$-torus is thus${\displaystyle n}$ choose${\displaystyle k}$.

More precisely, for adifferential manifoldM, one may equip it with some auxiliaryRiemannian metric. Then theLaplacian${\displaystyle \mathrm{\Delta }}$ is defined by

${\displaystyle \mathrm{\Delta }=d\delta +\delta d}$

with${\displaystyle d}$ theexterior derivative and${\displaystyle \delta }$ thecodifferential. The Laplacian is a homogeneous (ingrading)lineardifferential operator acting upon theexterior algebra ofdifferential forms: we can look at its action on each component of degree${\displaystyle k}$ separately.

If${\displaystyle M}$ iscompact andoriented, thedimension of thekernel of the Laplacian acting upon the space ofk-forms is then equal (byHodge theory) to that of the de Rham cohomology group in degree${\displaystyle k}$: the Laplacian picks out a uniqueharmonicform in each cohomology class ofclosed forms. In particular, the space of all harmonic${\displaystyle k}$-forms on${\displaystyle M}$ is isomorphic to${\displaystyle H^k(M;\mathbb{R}).}$ The dimension of each such space is finite, and is given by the${\displaystyle k}$-thBetti number.

Let${\displaystyle M}$ be acompactorientedRiemannian manifold. TheHodge decomposition states that any${\displaystyle k}$-form on${\displaystyle M}$ uniquely splits into the sum of threeL² components:

${\displaystyle \omega =\alpha +\beta +\gamma ,}$

where${\displaystyle \alpha }$ is exact,${\displaystyle \beta }$ is co-exact, and${\displaystyle \gamma }$ is harmonic.

One says that a form${\displaystyle \beta }$ is co-closed if${\displaystyle \delta \beta =0}$ and co-exact if${\displaystyle \beta =\delta \eta }$ for some form${\displaystyle \eta }$, and that${\displaystyle \gamma }$ is harmonic if the Laplacian is zero,${\displaystyle \mathrm{\Delta }\gamma =0}$. This follows by noting that exact and co-exact forms are orthogonal; the orthogonal complement then consists of forms that are both closed and co-closed: that is, of harmonic forms. Here, orthogonality is defined with respect to theL² inner product on${\displaystyle {\mathrm{\Omega }}^k(M)}$:

${\displaystyle (\alpha ,\beta )={\int }_M\alpha \wedge \star \beta .}$

By use ofSobolev spaces ordistributions, the decomposition can be extended for example to a complete (oriented or not) Riemannian manifold.^[6]

• Hodge theory
• Integration along fibers (for de Rham cohomology, thepushforward is given byintegration)
• Sheaf theory
• ${\displaystyle \mathrm{\partial }\overline{\mathrm{\partial }}}$-lemma for a refinement of exact differential forms in the case of compactKähler manifolds.

• Idea of the De Rham Cohomology inMathifold Project
• "De Rham cohomology",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Cohomology theories
• Differential forms

• Articles with short description
• Short description matches Wikidata