De Rham Cohomology

The exterior derivative $d$ from differential forms satisfies $d^2 = 0$. That single identity, which takes one line to verify, is the reason a branch of topology exists: any operator that squares to zero lets you form the quotient of the things it kills by the things that come from it, and the result is an invariant. For $d$ on differential forms the quotient is the de Rham cohomology of the manifold, and de Rham’s theorem says this algebraic object, built from calculus, is the same as the singular cohomology built from combinatorics.

The subject is the exact structural mirror of the measure theory in the algebraic topology page: there, chains and a boundary operator with $\partial^2 = 0$; here, forms and an exterior derivative with $d^2 = 0$. The duality between them is why integration of forms is possible at all; Stokes’ theorem is the statement that $\int_M d\omega$ depends only on the boundary of $M$, which is the statement that $d$ and $\partial$ are adjoint.

A form $\omega$ with $d\omega = 0$ is closed: nothing leaks. A form $\omega = d\eta$ is exact: it is the derivative of something, and so by Stokes its integral over any closed region is zero.

Every exact form is closed, because $d^2 = 0$. The interesting question is the converse: is every closed form exact? Locally yes; the Poincaré lemma below. Globally, sometimes no, and when it fails, the manifold has a hole. On $\mathbb{R}^2 \setminus \{0\}$, the form $\omega = \frac{-y\,dx + x\,dy}{x^2+y^2}$ is closed but not exact, and the obstruction is the puncture: integrating $\omega$ around a circle enclosing the origin gives $2\pi$, while Stokes would force $0$ if $\omega$ were $d\eta$. The integral sees the hole.

That is the entire subject in one example. Cohomology is the bookkeeping of which closed forms fail to be exact, and it is a topological invariant because the failure is caused by the shape of the space, not by the calculus.

Definition. Let $M$ be a smooth manifold. The space of closed $k$-forms is $Z^k(M) = \ker(d : \Omega^k(M) \to \Omega^{k+1}(M))$; the space of exact $k$-forms is $B^k(M) = \operatorname{im}(d : \Omega^{k-1}(M) \to \Omega^k(M))$ with $B^0 = 0$. The $k$-th de Rham cohomology group is

Proposition. $H^k_{\mathrm{dR}}(M)$ is a vector space over $\mathbb{R}$, and it is a diffeomorphism invariant: if $f : M \to N$ is a diffeomorphism then $f^* : H^k_{\mathrm{dR}}(N) \to H^k_{\mathrm{dR}}(M)$ is an isomorphism.

Proof. $\Omega^k(M)$ is a vector space and $d$ is linear, so kernels, images and quotients are vector spaces. For invariance: $d(f^*\omega) = f^*(d\omega)$ by naturality of $d$, so $f^*$ carries closed to closed and exact to exact - $f^*(d\eta) = d(f^*\eta)$; and therefore descends to the quotient. With $g = f^{-1}$, $g^* \circ f^* = (f \circ g)^* = \mathrm{id}$, so $f^*$ is an isomorphism. $\blacksquare$

Naturality of $d$ is doing all the work, and it is worth noticing: the invariance does not need $f$ to be a diffeomorphism, only smooth, so every smooth map induces a pullback on cohomology; which is what makes cohomology functorial rather than merely invariant.

Degree zero. $H^0_{\mathrm{dR}}(M)$ is the space of smooth functions with $df = 0$, i.e. locally constant functions. It is $\mathbb{R}^c$ where $c$ is the number of connected components; the zeroth invariant, again matching $H_0$ on the topology page.

Theorem (Poincaré lemma). On $\mathbb{R}^n$, every closed form of positive degree is exact. Equivalently $H^k_{\mathrm{dR}}(\mathbb{R}^n) = 0$ for $k \geq 1$, and $H^0 = \mathbb{R}$.

Proof. Define the radial contraction operator $K$ on forms by

where $\iota_x$ is contraction with the radial vector field $x$. A direct computation with Cartan’s magic formula $\mathcal{L}_x = d\iota_x + \iota_x d$ gives the homotopy identity

where $j_0^*$ pulls a form back along the inclusion of the origin; zero on positive-degree forms, evaluation at $0$ on functions. (The verification is mechanical: apply both sides to a $k$-vector at a point and compare terms, using the Leibniz rule for $\iota_x$ and the fundamental theorem of calculus on the $t$-integral.)

Now let $\omega$ be closed with $k \geq 1$. Applying the identity,

so $\omega = d(K\omega)$ is exact, with the primitive given explicitly by $K\omega$. $\blacksquare$

The proof produces the primitive, not merely its existence, which is why the radial contraction is worth writing out. And the identity $dK + Kd = \mathrm{id} - j_0^*$ is the general shape of every cohomology argument: a chain homotopy between the identity and something degenerate, giving equality in cohomology where the two differ at chain level.

Where it fails. The lemma needs a star-shaped domain, so that the radial contraction is defined. On an annulus there is no contraction to a point, and the closed-but-not-exact form above is the obstruction. Cohomology measures exactly the failure of the Poincaré lemma.

Proposition. On $M = \mathbb{R}^2 \setminus \{0\}$, the form

is closed but not exact. Hence $H^1_{\mathrm{dR}}(M) \neq 0$.

Proof. Closed. Writing $r^2 = x^2+y^2$, differentiate and collect:

The middle two terms vanish because $dx \wedge dx = dy \wedge dy = 0$, and the bracket is

Not exact. Parametrise the unit circle by $\gamma(t) = (\cos t, \sin t)$:

If $\omega = d\eta$ then by Stokes $\int_\gamma \omega = \int_\gamma d\eta = \int_{\partial D} d\eta = 0$, where $D$ is the unit disc; but $\gamma = \partial D$, and the integral is $2\pi \neq 0$. Contradiction. $\blacksquare$

Note what the computation does not do: it does not compute $H^1$ exactly, only show it is nonzero. Doing better requires either the full de Rham theorem or a Mayer–Vietoris argument, and the answer is $H^1_{\mathrm{dR}}(M) = \mathbb{R}$, generated by the class of $\omega$.

The angle subtlety. The form $\omega$ is $d\theta$ locally, where $\theta$ is the polar angle; but $\theta$ is not a globally defined smooth function on $M$, which is exactly why $\omega$ can be closed without being exact. The obstruction to writing down the primitive globally is the topology.

Theorem. If $M$ is a connected, orientable, compact $n$-manifold, then $H^n_{\mathrm{dR}}(M) \cong \mathbb{R}$.

Proof sketch. A volume form exists exactly when $M$ is orientable, and two volume forms with the same total integral differ by the derivative of an $(n-1)$-form; this is the statement that the “total integral” map $\Omega^n(M) \to \mathbb{R}$, $\omega \mapsto \int_M \omega$, has kernel exactly the exact forms, which follows from the fact that on a connected manifold any two points can be joined and the integral is unchanged by moving the support of the primitive along the path. Stokes says $\int_M d\eta = 0$ for a closed $M$, so $B^n \subseteq \ker$; the content is the converse. $\blacksquare$

For non-orientable connected $M$, $H^n = 0$: there is no volume form to integrate, and the top class vanishes. So orientability is a cohomological property, which is why it can be detected algebraically.

The Euler characteristic link from Gauss–Bonnet comes from the alternating sum of the dimensions of these spaces:

which is de Rham’s theorem plus the definition of Betti numbers, and it is why $\chi$ can be computed either by triangulating or by solving PDE.

Theorem (de Rham, 1931). The natural map from de Rham cohomology to singular cohomology with real coefficients is an isomorphism:

What it does. It says calculus sees topology. The closed form $(-y\,dx + x\,dy)/r^2$ is a smooth object, defined with partial derivatives, and it detects a puncture. Every invariant of singular cohomology; Betti numbers, cup products, Poincaré duality; is available in the language of forms, where it can be computed by integration and PDE.

What it does not. It says nothing over $\mathbb{Z}$: torsion in singular cohomology is invisible to de Rham theory, since $H^k_{\mathrm{dR}}$ is a real vector space and has none. The standard example is $\mathbb{R}P^2$, whose $H^1_{\mathrm{sing}}(\cdot; \mathbb{Z}) = \mathbb{Z}/2$ collapses to $0$ over $\mathbb{R}$. So de Rham theory sees the free part and is blind to torsion; and a question whose answer involves torsion cannot be asked in the language of differential forms.

• Concluding “closed hence exact” from the Poincaré lemma. The lemma is local, and on star-shaped domains. Applying it to $\mathbb{R}^2\setminus\{0\}$ is the canonical error, and the angle form is the canonical counterexample.
• Verifying $d\omega = 0$ sloppily. Two of the four terms of $d\omega$ for a 1-form in two variables vanish because $dx\wedge dx = 0$; forgetting the signs in the remaining pair gives the wrong conclusion. The computation in §3 keeps all four terms visible.
• Citing Stokes where Stokes is not licensed. In §3 the contradiction uses Stokes on the punctured disc, which is not compact; the correct statement applies to the unit disc, with the circle as boundary, and needs $\eta$ to be defined on all of $D$. Since $M$ excludes the origin, $\eta$ would be, and the argument is valid; but writing Stokes on $M \setminus \{0\}$ directly would be wrong.
• Confusing $H^0$ with functions. $H^0$ is locally constant functions, not all smooth functions. On a connected manifold it is $\mathbb{R}$, on a disconnected one $\mathbb{R}^c$.
• Expecting de Rham classes to detect torsion. They cannot, by construction: real vector spaces have no torsion. $\mathbb{R}P^2$ is the standard warning.
• Treating the de Rham theorem as “de Rham cohomology is singular cohomology” without the coefficient group. The isomorphism is with $\mathbb{R}$ coefficients only.

• Differential Forms; the exterior derivative and Stokes’ theorem, both used from the first line.
• Smooth Manifolds; the definition of the underlying object and of pullback.
• The Gauss–Bonnet Theorem; $\chi$ as the alternating sum of Betti numbers.
• Introduction to Algebraic Topology; the chain-level mirror: $\partial^2 = 0$ and singular homology, of which de Rham theory is the dual. Its §9.8 proves the Mayer–Vietoris sequence and computes $H^*(S^2)$ and $H^*(\Sigma_2)$ with it, which is the computation this page’s §3 shows is needed before $H^1$ can be pinned down.
• Hodge Theory; every de Rham class has a unique harmonic representative; the dimension of the harmonic forms is the Betti number.
• The Fundamental Theorems; the self-adjoint operator theory that the Hodge Laplacian rests on.
• Applications of Differential Geometry; Maxwell’s equations in the language of $d$, where $dF = 0$ and $d{*F} = J$ are the closed-form conditions.

• Applications
• Curvature
• Differential Forms
• Elliptic Regularity