nLab de Rham theorem

Idea

On cohomology

The de Rham theorem (named after Georges de Rham) asserts that the de Rham cohomology $H^n_{dR}(X)$ of a smooth manifold $X$ (without boundary) is isomorphic to the real cohomology $H^n(X, \mathbb{R})$, 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 $\mathbb{R}_c$ for

• the abelian group $\mathbb{R}$

• regarded not as a Lie group with the standard manifold structure on $\mathbb{R}$ but as a topologically discrete group on the underlying set of $\mathbb{R}$

• and then regarded as a sheaf on $X$: the constant sheaf that sends connected $U \subset X$ to the set of constant maps $U \to \mathbb{R}$.

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

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

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

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

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

Then we have:

• “ordinary” $\mathbb{R}$-valued cohomology of $X$ is the abelian sheaf cohomology with coefficients in $\mathbf{B}^n \mathbb{R}_c$.

• de Rham cohomology of $X$ is the abelian sheaf cohomology with coefficients in $\bar \mathbf{B}^n \mathbb{R}$ (this is semi-obvious, requires a bit more discussion).

• the Poincare lemma says that every closed differential form is locally exact, and hence there is a quasi-isomorphism of chain complexes of sheaves

  $$\mathbf{B}^n \mathbb{R}_c \stackrel{\simeq}{\to} \bar \mathbf{B}^n \mathbb{R}$$

  given by injecting for each $U \subset X$ the set $\mathbb{R}$ as the constant functions into $C^\infty(U,\mathbb{R})$.

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

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

…

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 $X$ a smooth manifold there is an equivalence of A-infinity algebras

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

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 $\mathcal{T}$ modelling the axioms of synthetic differential geometry.

Specifically

• the internal singular chain complex in $\mathcal{T}$ is given as the $R$-linear dual of the free internal $R$-module on the internal hom objects $[\Delta^n,X]$, where $R$ is the internal incarnation of the real numbers;

• the de Rham complex is given by differential forms in synthetic differential geometry.

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

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

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

This is discussed in chapter IV of

• Ieke Moerdijk, Gonzalo Reyes, Models for Smooth Infinitesimal Analysis for the toposes $\mathcal{T} = \mathcal{G}, \mathcal{F}$ (see models).

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

• Patrick Iglesias-Zemmour, De Rham calculus (pdf)

Related concepts

• PL de Rham theorem

• equivariant de Rham theorem

• equivariant PL de Rham theorem

• Poincaré lemma

• Stokes theorem

• de Rham theorem

  • kernel of integration is the exact differential forms

• Dolbeault theorem

• Hodge theorem

References

On cohomology

Textbook accounts:

• M M Postnikov, Differentiable manifolds, vol. III of: Lectures on geometry,

• Raoul Bott, Loring Tu, Thm. 8.9 & Prop. 10.6 of: Differential Forms in Algebraic Topology, Springer (1982)

• Jean Gallier, Jocelyn Quaintance, Ch. 9 in: Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, World Scientific (2022) [doi:10.1142/12495, webpage]

Exposition:

• Arne Lorenz, Abstract de Rham theorem, pdf slides

In the context of rational homotopy theory:

• Yves Félix, Stephen Halperin, Jean-Claude Thomas, Theorem 10.15 in: Rational Homotopy Theory, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 (doi:10.1007/978-1-4613-0105-9)

In analytic geometry:

• M. E. Herrera, De Rham theorems on semianalytic sets, Bull. Amer. Math. Soc. 7 3 (1967) 414–418, doi, MR214094

In the generality of diffeological spaces:

• Patrick Iglesias-Zemmour, Une cohomologie de Čech pour les espaces differentiables et sa relation a la cohomologie de De Rham (1988) [pdf, pdf]

• Patrick Iglesias-Zemmour, Čech–De Rham bicomplex in diffeology, Israel Journal of Mathematics (2023) 1–38 [doi:10.1007/s11856-023-2486-8]

• Emilio Minichiello, The Diffeological Čech-de Rham Obstruction [arXiv:2401.09400]

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

• Victor Gugenheim, On Chen’s iterated integrals , Illinois J. Math. Volume 21, Issue 3 (1977), 703-715.

proven using Chen’s iterated integrals.

A review is in section 3 of

• Camilo Arias Abad, Florian Schätz, The $A_\infty$ de Rham theorem and integration of representations up to homotopy (arXiv:1011.4693)