de Rham theorem




Special and general types

Special notions


Extra structure



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories




On cohomology

The de Rham theorem (named after Georges de Rham) asserts that the de Rham cohomology H dR n(X)H^n_{dR}(X) of a smooth manifold XX (without boundary) is isomorphic to the real cohomology H n(X,)H^n(X, \mathbb{R}), hence its ordinary cohomology with real number coefficients, as computed for instance by the singular or Čech cohomology with real coefficients.

The theorem has several dozens of different proofs. For example in the Čech approach one can make a double complex whose first row is the Čech complex of a covering and first column is the de Rham complex and other entries are mixed and use spectral sequence argument (see the textbook of Bott and Tu, or the geometry lectures book by Postnikov, semester III).

This is maybe best formulated, understood and proven in the context of abelian sheaf cohomology:

Write c\mathbb{R}_c for

Write B n c\mathbf{B}^n \mathbb{R}_c for the corresponding Eilenberg-MacLane object in chain complexes of sheaves of abelian groups: this is the complex of sheaves with c\mathbb{R}_c in degree nn:

B n c=(0 c00). \mathbf{B}^n \mathbb{R}_c = (\cdots \to 0 \to \mathbb{R}_c \to 0 \to \cdots \to 0) \,.

Next, write B¯ n\bar \mathbf{B}^n \mathbb{R} (without the subscript cc!) for the Deligne complex for \mathbb{R}

B¯ n=(C (,)d dRΩ 1()d dRΩ 2()d dRd dRΩ closed n()). \bar \mathbf{B}^n \mathbb{R} = (C^\infty(-,\mathbb{R}) \stackrel{d_{dR}}{\to} \Omega^1(-) \stackrel{d_{dR}}{\to} \Omega^2(-) \stackrel{d_{dR}}{\to} \cdots \stackrel{d_{dR}}{\to} \Omega^n_{closed}(-)) \,.

(The notation here is borrowed from that used at motivation for sheaves, cohomology and higher stacks: we can think of B¯ n\bar \mathbf{B}^n \mathbb{R} as a differential refinement of the object B n c\mathbf{B}^n \mathbb{R}_c).

Then we have:

It is this quasi-isomorphism of coefficient objects that induces the de Rham isomorphism of abelian sheaf cohomology groups, which is ordinarily written as

H n(X,)H dR n(X). H^n(X,\mathbb{R}) \simeq H^n_{dR}(X) \,.

On cochains

The equivalence on cohomology asserted by the de Rham theorem is but a decategorification of a more refined statement: a quasi-isomorphism of cochain complexes. This even respects the product structure:

for XX a smooth manifold there is an equivalence of A-infinity algebras

(Ω (X),d dR)(C(X),) (\Omega^\bullet(X), d_{dR}) \stackrel{\simeq}{\to} (C(X), \cup)

between the de Rham complex and the collection of singular cochains equipped with the cup product.

This is due to (Gugenheim, 1977).

Furthermore, the E-infinity algebra structure on differential forms (trivially induced by the commutative dga? structure) and singular cochains (as witnessed by the action of the sequence operad of McClure and Smith on singular cochains) is also preserved.

Synthetic version

The de Rham theorem also holds internally in the context of suitable smooth toposes 𝒯\mathcal{T} modelling the axioms of synthetic differential geometry.


The de Rham theorem in 𝒯\mathcal{T} then asserts that for XX a manifold regarded as an object in the well-adapted smooth topos 𝒯\mathcal{T} the morphism

:H p(X)H p(X,R) * \int : H^p(X) \to H_p(X,R)^*

in 𝒯\mathcal{T} is an isomorphism for all pp \in \mathbb{N}. This implies the standard (external) de Rham theorem.

This is discussed in chapter IV of

A little bit a long these lines for diffeological spaces is also in


On cohomology

Standard textbook references include

In the broader context of rational homotopy theory:

In analytic geometry also

On cocycles

The refinement of the de Rham theorem from an isomorphism of cohomology groups to an equivalence of A-∞ algebras of cochains and forms was first stated in

proven using Chen’s iterated integrals.

A review is in section 3 of