de Rham Cohomology

de Rham Cohomology: Topology Through Analysis

When Does Closed Not Mean Exact?

From $d^2 = 0$ it follows: exact form ($\omega = d\eta$) $\rightarrow$ closed ($d\omega = 0$). But the converse does not always hold: closed $\nrightarrow$ exact. This "gap" is connected with the topology of the manifold—the presence of "holes."

Intuition: on the punctured plane $\mathbb{R}^2 \setminus {0}$ the form $\omega = \frac{x,dy - y,dx}{x^2 + y^2}$ is closed ($d\omega = 0$), but $\oint_{|r| = 1} \omega = 2\pi \ne 0$. If $\omega = df$, then $\oint \omega = 0$. Therefore, $\omega$ is not exact—the "hole" at the origin obstructs exactness.

This is a deep idea: topological properties of space (holes, "handles") are detected by analytic means (integration of forms).

de Rham Groups

For $k \ge 0$ define:

$Z^k(M) = {\text{closed } k\text{-forms: } d\omega = 0}$—"kernel of $d

$B^k(M) = {\text{exact } k\text{-forms: } \omega = d\eta}$—"image of $d

The $k$-th de Rham cohomology group:

$H^k(M) = Z^k(M) / B^k(M)$

Elements of $H^k$ are equivalence classes of forms up to addition of an exact form. If $\omega_1$ and $\omega_2$ differ by an exact form, they represent the same class.

Numerical Examples

$H^0(M) = \mathbb{R}^{\pi_0(M)}$ ($\pi_0$ is the number of connected components). For connected $M$: $H^0(M) = \mathbb{R}$. Explanation: a 0-form $f$ is closed if and only if $df = 0$, i.e., $f$ is constant. There are no exact 0-forms (no forms of degree $-1$). Thus, $H^0 = {$constants$} = \mathbb{R}$ for connected $M$.

$H^1(S^1) = \mathbb{R}$: The 1-form $d\theta$ is closed ($d(d\theta) = 0$), but not exact ($\oint d\theta = 2\pi \ne 0$). All closed 1-forms on $S^1$ are $a,d\theta + df$ for $a \in \mathbb{R}$. $H^1 = \mathbb{R}$, generated by the class $[d\theta]$.

$H^1(\mathbb{R}^n) = 0$: The Poincaré lemma—everything is contractible.

$H^k(S^n) = \mathbb{R}$ for $k = 0, n$ and $0$ otherwise: The "hole" of the sphere is in itself.

$H^1(T^2) = \mathbb{R}^2$: The torus has two independent cycles (meridian and parallel) $\rightarrow$ two independent classes. $H^2(T^2) = \mathbb{R}$ (the volume of the torus).

de Rham's Theorem

de Rham's Theorem (1931): $H^k_{dR}(M) \cong H^k_{sing}(M; \mathbb{R})$ (singular cohomology). de Rham groups are invariants of the smooth structure, but coincide with purely topological invariants.

Betti numbers: $\beta_k = \dim H^k(M; \mathbb{R})$. $\beta_0 =$ number of components, $\beta_1 =$ "number of holes", $\beta_2 =$ "number of voids", ...

Euler characteristic: $\chi(M) = \sum_k (-1)^k \beta_k$. For the sphere $S^2$: $\beta_0=1$, $\beta_1=0$, $\beta_2=1$, $\chi = 1-0+1 = 2$ ✓. For the torus $T^2$: $\beta_0=1$, $\beta_1=2$, $\beta_2=1$, $\chi = 1-2+1 = 0$ ✓.

Characteristic Classes

Characteristic classes are "global invariants of bundles," computable through local curvature.

Chern class $c_1(L)$: For a complex line bundle $L$ with curvature $\Omega$: $c_1(L) = [\Omega/(2\pi i)] \in H^2(M; \mathbb{Z})$. If $M$ is a complex manifold, $c_1 \in \mathbb{Z}$—the first Chern number.

Gauss–Bonnet theorem as a special case: For the tangent bundle of a surface: $c_1(TM) = \chi(M)/2$. The formula $\int_M K,dA = 2\pi\chi(M)$ is the integral of the form representing $c_1$.

Atiyah–Singer theorem (1963): For an elliptic operator $D$ on a closed manifold: $\operatorname{index}(D) = \int_M \operatorname{ch}(\sigma_D), \operatorname{td}(TM)$. The index (an integral analytic invariant) is expressed via characteristic classes. This is a generalization of Gauss–Bonnet to arbitrary operators—one of the great theorems of the 20th century.

de Rham Cohomology in Physics

Maxwell's equations are naturally reformulated through differential forms: the electromagnetic field $F = dA$ is an exact 2-form ($A$ is a 1-form potential). The equation $dF = 0$ encodes $\nabla\cdot B = 0$ and Faraday's law. The class $[F] \in H^2(M)$ fixes the topological obstruction for a global potential. On a manifold with a "hole" (solenoid) the potential $A$ is ambiguous—hence the Aharonov–Bohm effect: the interference of electrons encircling a solenoid depends on $\oint A\cdot dl$, even though $F = 0$ outside. This is a direct physical manifestation of cohomology.

de Rham Cohomology in Field Physics and Topological Data Analysis

de Rham cohomology classifies global obstructions to integrability and has wide applications. In electrodynamics cohomology explains why a magnetic monopole can exist only on manifolds with nontrivial $H^2$: violation of $d(\star F) = 0$ outside the monopole means that $\star F$ is not exact, and thus its cohomology class is nonzero. This mathematical condition induces quantization of magnetic charge (the Dirac condition). In quantum mechanics the Berry phase describes the geometric phase upon adiabatic traversal of a closed path in parameter space: it is expressed through the curvature of a "bundle" over parameter space and is a cohomological invariant. The integer quantum Hall effect, discovered in 1980 and explained topologically by Thouless (Nobel Prize 2016), is described by the Chern number—an element of $H^2$(zone space). In topological data analysis (TDA), persistent homology and cohomology analyze "holes" in point clouds: the method is used in neuroscience data analysis, materials science, and bioinformatics to detect structure in high-dimensional data.

In supersymmetric field theory, the de Rham–Hodge theorem—the decomposition of the space of forms into harmonic, exact, and coexact—corresponds to the structure of supercharges $Q$ and $Q^\dagger$: Betti numbers are computed as the index of the Dirac–Hodge operator via supersymmetric quantum mechanics. This is Witten's theorem, which provided a new topological approach to manifold invariants and unified the physics of supersymmetry with Morse theory.

Exercise: (a) For $M = S^1 \times S^1$ (the torus): compute all $H^k(M)$ and Betti numbers. (b) The form $\omega = \frac{x,dy - y,dx}{x^2 + y^2}$ on $\mathbb{R}^2 \setminus {0}$: find $[\omega] \in H^1(\mathbb{R}^2 \setminus {0})$. Does $\omega$ generate the entire group $H^1$? (c) What is $H^k(S^n)$? Compute for $n=3$ and interpret physically (connection with magnetic monopoles and topological charge).