Hodge Theory

De Rham cohomology is a quotient: closed forms modulo exact forms. Quotients are algebraically convenient and analytically awkward, because a cohomology class has no distinguished representative; any element of the class will do, and nothing singles one out. Hodge theory supplies the missing selection principle: on a Riemannian manifold, each class contains exactly one harmonic form, the one solving a PDE.

That turns a topological invariant into a statement about analysis. The dimension of $H^k_{\mathrm{dR}}(M)$ equals the dimension of the space of harmonic $k$-forms, which is the eigenspace of a self-adjoint elliptic operator at eigenvalue zero - and so everything proved about self-adjoint operators in the fundamental theorems applies to topology. This is the cleanest instance of a pattern that runs through all of geometry: to study topology, build an elliptic operator and compute its kernel.

Picture the space of all $k$-forms as a big vector space, chopped into the subspace of exact forms $B^k$ sitting inside the closed forms $Z^k$. A cohomology class is a coset; a whole affine copy of $B^k$. The Riemannian metric gives the ambient space an orthogonal complement, and the Hodge theorem says: each coset meets the complement of $B^k$ in exactly one point. That point is the harmonic representative.

Why is the complement the right choice? Because the complement of the exact forms is the kernel of the Laplacian, and kernels are computable. So the selection principle is not arbitrary: it is the orthogonal projection, the one choice that minimises the norm among all representatives of the class. The harmonic representative of a cohomology class is its least-squares solution, which is why the theory connects to the variational methods used everywhere else in geometry.

Let $M$ be a compact, orientable, Riemannian $n$-manifold, with metric $g$. The metric induces:

• the Hodge star $*: \Omega^k(M) \to \Omega^{n-k}(M)$, defined by $\alpha \wedge *\beta = \langle \alpha, \beta\rangle\, dV$, satisfying $*^2 = (-1)^{k(n-k)}$ on $k$-forms;
• the codifferential $d^* = (-1)^{n(k+1)+1} * \, d \, *$ on $\Omega^k$;
• the Hodge Laplacian

Proposition. $\Delta$ is self-adjoint and non-negative: $\langle \Delta\alpha, \alpha\rangle = \|d\alpha\|^2 + \|d^*\alpha\|^2 \geq 0$.

Proof. Both $d$ and $d^*$ are adjoints of one another by construction, and $d^2 = (d^*)^2 = 0$. Then

using $d^2 = 0$ and $(d^*)^2 = 0$ in the cross terms: they are $\langle d^*\alpha, d\alpha\rangle$ twice with opposite pairing, cancelling. The right side is a sum of squared norms. $\blacksquare$

This is where self-adjointness earns its keep, and it is worth comparing with Theorem 1 of the spectral theory page: there, self-adjointness gave real spectrum; here it gives an orthogonal decomposition of the form space, which is what the theorem below needs. The estimate $\langle\Delta\alpha,\alpha\rangle = \|d\alpha\|^2 + \|d^*\alpha\|^2$ is the same shape as $\|(T-z)x\|^2 = \|(T-u)x\|^2 + v^2\|x\|^2$, and it plays the same role: it turns an algebraic condition into a norm statement.

Definition. $\alpha$ is harmonic if $\Delta\alpha = 0$. The space of harmonic $k$-forms is $\mathcal{H}^k(M) = \ker\Delta$.

Corollary. $\alpha$ is harmonic iff $d\alpha = 0$ and $d^*\alpha = 0$.

Proof. $\Delta\alpha = 0$ forces $\|d\alpha\|^2 + \|d^*\alpha\|^2 = 0$, and a sum of squares vanishes termwise. The converse is immediate. $\blacksquare$

So a harmonic form is one that is simultaneously closed and co-closed; it leaks in neither direction. On a compact manifold, harmonicity is two first-order conditions rather than one second-order equation, which is why the theory is computable.

Theorem (Hodge decomposition). For each $k$ there is an orthogonal direct sum

Proof sketch. $\Delta$ is self-adjoint and elliptic. Self-adjointness gives $(\operatorname{im}\Delta)^\perp = \ker\Delta = \mathcal{H}^k$ on a compact manifold, and elliptic regularity; the analytic input this page uses rather than proves; says $\operatorname{im}\Delta$ is closed and that elements of $\ker\Delta$ are smooth. Hence the first decomposition. For the second, $d\Omega^{k-1}$ and $d^*\Omega^{k+1}$ are orthogonal because $\langle d\alpha, d^*\beta\rangle = \langle d^2\alpha, \beta\rangle = 0$, and their sum is the orthogonal complement of $\mathcal{H}^k$ inside $\Delta\Omega^k$. $\blacksquare$

The analytic input; elliptic regularity, which says the kernel of an elliptic operator consists of smooth sections; is genuinely deep, and this page uses it as a black box. Flagging that is part of stating what has been proved: the algebra of the decomposition is elementary, and the regularity theorem is not. That theorem is proved in Elliptic Regularity - Weyl’s lemma and the difference-quotient interior regularity for Poisson’s equation.

Theorem (Hodge). Every de Rham cohomology class contains exactly one harmonic representative. Consequently

Proof. Take $[\omega] \in H^k_{\mathrm{dR}}$ with $\omega$ closed, and use the decomposition to write $\omega = d\alpha + d^*\beta + h$ with $h$ harmonic. Since $d\omega = 0$ and $d(d\alpha) = 0$:

and pairing with $\beta$ gives $0 = \langle d d^*\beta, \beta\rangle = \|d^*\beta\|^2$, so $d^*\beta = 0$ and the $d^*\beta$ term vanishes. Hence $\omega = d\alpha + h$, so $[\omega] = [h]$ - every class has a harmonic representative.

For uniqueness, suppose $h_1, h_2$ are harmonic with $[h_1] = [h_2]$. Then $h_1 - h_2 = d\alpha$ is both harmonic and exact, so

and $d(h_1 - h_2) = 0$ by harmonicity, so the norm is zero and $h_1 = h_2$. $\blacksquare$

The proof is short because the work is in the decomposition, and the decomposition is short because the work is in elliptic regularity. That is the honest shape of the subject: the statement is topological, the machinery is PDE, and the algebra in between is three lines.

Consequence: Betti numbers are analytic. $b_k = \dim \ker\Delta$ on $k$-forms, so the Betti numbers of a manifold are the multiplicities of the zero eigenvalue of an elliptic operator. Since eigenvalue multiplicities cannot change under continuous deformation of the metric, this gives the analytic proof that the Betti numbers are topological; no triangulation required.

Consequence: Poincaré duality, analytically. The Hodge star maps harmonic $k$-forms to harmonic $(n-k)$-forms, so $*: \mathcal{H}^k \to \mathcal{H}^{n-k}$ is an isomorphism and $b_k = b_{n-k}$. Poincaré duality, normally proved by dual cell structures, here follows from $*^2 = 1$.

The flat torus $T^n$. Take $M = \mathbb{R}^n/\mathbb{Z}^n$ with the flat metric. The $k$-forms $dx_{i_1}\wedge\dots\wedge dx_{i_k}$ have constant coefficients, and $\Delta$ on constant-coefficient forms is zero. So $\dim\mathcal{H}^k = \binom{n}{k}$ and every Betti number is binomial; computed without solving a PDE, because on a flat torus every form with constant coefficients is harmonic and there are no others. The general theorem says any metric on $T^n$ gives the same answer, which is the content: the Laplacian depends on the metric and $b_k$ does not.

The circle. $\mathcal{H}^0 =$ constants, $\mathcal{H}^1 =$ multiples of $dx$. So $b_0 = b_1 = 1$, matching $S^1$‘s two invariants from algebraic topology: one connected component, one hole.

The sphere $S^2$. $\mathcal{H}^0$ and $\mathcal{H}^2$ are one-dimensional (volume form) and $\mathcal{H}^1 = 0$: there are no harmonic 1-forms, because $S^2$ has no tunnels. The vanishing is the analytic shadow of simple connectedness, and it is the same statement as $H^1_{\mathrm{dR}}(S^2) = 0$ from the Mayer–Vietoris computation on the topology page.

It requires a metric, so it sees nothing metric-independent directly; the harmonic representative depends on $g$, and only its cohomology class does not. That is a feature, not a defect, but it means Hodge theory answers “is there a canonical representative” at the cost of choosing a metric first.

It sees no torsion, for the same reason de Rham theory does not: everything is a real vector space. $\mathbb{R}P^2$ remains the standing warning.

And on non-compact manifolds the theory changes character entirely: compactness is what makes $\ker\Delta$ finite-dimensional, and without it the space of harmonic forms can be infinite-dimensional and the theorem fails. $L^2$ Hodge theory is the repair, and it is a genuinely different subject.

• Reading $\Delta$ as the rough Laplacian. The Hodge Laplacian is $dd^* + d^*d$, not the connection Laplacian $\nabla^*\nabla$; they differ by a curvature term (the Weitzenböck formula). On functions they agree, which is why the distinction is invisible in elementary Riemannian geometry and sudden here.
• Assuming harmonicity is a second-order equation with a first-order solution. It is two first-order conditions, $d\alpha = 0$ and $d^*\alpha = 0$, on a compact manifold; and on a non-compact one that equivalence fails without an integrability condition.
• Applying the decomposition without compactness. On a non-compact manifold $\operatorname{im}\Delta$ need not be closed, and the Hodge theorem fails. Compactness is used in every step after the definition.
• Forgetting that the harmonic representative depends on the metric. Only its class is canonical. Two metrics give two different harmonic forms for the same class, and comparing them is a question about the metrics, not the topology.
• Expecting to compute $\mathcal{H}^k$ without solving a PDE. The theorem says the dimension is topological; it does not make the forms explicit. On a generic metric the harmonic forms are as hard to write down as eigenfunctions are.

• De Rham Cohomology; the quotient that Hodge theory picks representatives for.
• Differential Forms; the exterior derivative and the Hodge star.
• Riemannian Geometry; the metric inducing $*$ and $d^*$.
• The Fundamental Theorems; self-adjointness and the orthogonal decomposition used in §2.
• Introduction to Algebraic Topology; $b_k$, Poincaré duality, and the Mayer–Vietoris computation of $H^*(S^2)$ recovered analytically in §4.
• The Gauss–Bonnet Theorem; $\chi = \sum(-1)^k \dim\mathcal{H}^k$, the alternating sum of kernel dimensions.
• Quantum Mechanics; the same self-adjoint operator theory, with the harmonic oscillator as the model example.

• Applications
• Curvature
• De Rham Cohomology
• Differential Forms