Calculus

De Rham Cohomology

Цели урока

  • Understand de Rham cohomology as the gap between closed and exact forms
  • Know the de Rham theorem about the isomorphism with singular cohomology
  • Master Hodge's theorem on harmonic representatives and the Hodge decomposition
  • Compute Betti numbers via the Kunneth formula and the Mayer-Vietoris sequence

Предварительные знания

  • Integration on manifolds
  • Hodge operator
  • Exterior derivative d^2 = 0
  • Integration on manifolds

How do you count the 'holes' in a space without relying on intuition, and why do smooth equations (forms) give the same answer as combinatorics (simplices)?

  • Topological data analysis (TDA): persistent homology computes Betti-number analogs for molecular structures in AlphaFold2 and computational chemistry
  • Quantum physics: topological quantum numbers (Chern-Simons charges) are cohomological invariants robust to noise
  • Neural potential-field networks: a conservative field (rot F = 0) is the exactness condition of a 1-form, encoding H^1 = 0
  • Gudhi / Ripser (Python): persistent Betti numbers for 3D objects, images, and time series in data science

De Rham and the bridge between geometry and topology

Georges de Rham (1903-1990) was a Swiss mathematician, a student of Lefschetz and Cartan. His 1931 thesis 'Sur l'Analysis situs des varietes a n dimensions' proved that differential forms and singular chains compute the same invariant. The theorem became the bridge: Poincare introduced Betti numbers combinatorially in 1895; de Rham showed they could be computed analytically through d. Hodge (1941) added a third viewpoint: harmonic forms. Three languages, one mathematics. Today de Rham cohomology sits at the foundation of high-energy physics, mirror symmetry in string theory, and topological quantum field theory.

Closed and exact forms: de Rham cohomology

1931. Georges de Rham defends his thesis in Paris. His result: smooth geometry and topology speak the same language. The angular form dtheta = (x dy - y dx)/(x^2 + y^2) is closed (d(dtheta) = 0) but not exact on R^2 minus the origin. Its integral over S^1 equals 2pi - nonzero. That fact encodes the 'hole' in the space. De Rham made it precise.

Betti numbers of standard spaces: R^n: beta_0 = 1, beta_k = 0 for k > 0. S^n: beta_0 = beta_n = 1, the rest 0. T^2: beta_0 = 1, beta_1 = 2, beta_2 = 1. CP^2: beta_0 = beta_2 = beta_4 = 1. Betti numbers are topological invariants.

H^1_dR(S^1) = R. Which form generates this class?

The angular form dtheta is closed: d(dtheta) = 0. But int_{S^1} dtheta = 2pi != 0, so dtheta != df for any single-valued function f. That is what encodes H^1(S^1) = R.

Hodge's theorem and harmonic representatives

Every cohomology class contains infinitely many forms. Hodge's theorem singles out a canonical representative: the harmonic form of minimum norm. Analogy: among all paths between two points on a sphere, the geodesic stands out. Among all forms in a class, the harmonic one stands out. Minimum norm = Laplace equation.

On the torus T^n the harmonic forms are constants: omega = sum c_I dx^I. For T^2: harmonic 0-forms = constants (1-dimensional). Harmonic 1-forms = a dx + b dy (2-dimensional = beta_1 = 2). Harmonic 2-forms = c dx wedge dy (1-dimensional).

Harmonic functions and Hodge's theorem

Classical harmonic functions as a special case

On an open region in R^n: H^0 = ker(Delta_0) = {f: Delta f = 0} - harmonic functions in the classical sense. Hodge's theorem on a compact M without boundary: dim(H^0) = number of connected components of M (a constant function on each). For S^n: H^0 = R (one component). For S^1 union S^1: H^0 = R^2.

Hodge decomposition: Omega^k = H^k + im(d) + im(d*). For omega = d alpha + d*(beta) + gamma (gamma harmonic), which piece is which?

The Hodge decomposition is orthogonal: omega = gamma + d(alpha) + d*(beta), with the three pieces mutually perpendicular. omega is closed iff d*(beta) = 0 iff omega = exact + harmonic component.

Kunneth formula and Mayer-Vietoris

How do you compute cohomology of complicated spaces from cohomology of simple parts? Two tools: the Kunneth formula (for products) and the Mayer-Vietoris sequence (for unions). These are algebraic machines that make cohomology computable. The same machines drive homology in topological data analysis (persistent homology), which AlphaFold uses to analyze molecular structures.

Cohomology of the torus via the Kunneth formula

T^2 = S^1 x S^1

H^0(S^1) = R, H^1(S^1) = R. By the Kunneth formula: H^0(T^2) = H^0(S^1) tensor H^0(S^1) = R. H^1(T^2) = H^0 x H^1 + H^1 x H^0 = R + R = R^2. H^2(T^2) = H^1 x H^1 = R. Betti numbers: (1, 2, 1). Euler characteristic: 1 - 2 + 1 = 0. The torus is topologically 'neutral'.

De Rham cohomology over R cannot distinguish, for example, the sphere S^2 from the projective plane RP^2 by integer invariants. Integer invariants need integer cohomology - de Rham cohomology only gives real Betti numbers.

H^1_dR(T^2) = R^2. This means the torus has:

beta_1 = 2 for T^2: two independent 1-cycles (meridian and longitude). Each cycle generates its own class in H^1_dR. Their sum spans a two-dimensional space.

Cohomology in physics and data topology

De Rham cohomology is not an abstraction. In physics: topological quantum numbers for charges are governed by the first cohomology group. In data topology: persistent homology builds analogs of Betti numbers for point clouds. AlphaFold2 uses geometric invariants of molecules - discrete versions of Betti numbers. The firm Ayasdi built financial tools on topological data analysis.

De Rham's theorem in action in ML: in point-cloud classification (geometric ML) Betti numbers are shape invariants. Persistent homology builds them for noisy data. The Gudhi library (C++/Python) computes persistent Betti numbers for molecules, images, and 3D objects. That is the tool used to search for new materials in computational chemistry.

Topological invariants are robust to noise, unlike geometric ones (lengths, angles). That is why persistent homology is popular in data science: the shape of data (number of holes) does not change under small deformations. Robust to noise by design.

Morse inequality: c_1 >= beta_1. For the torus T^2 (beta_1 = 2) this means:

Morse inequality c_k >= beta_k: you cannot build a function with fewer critical points than beta_k. For T^2: c_0 >= 1, c_1 >= 2, c_2 >= 1. Minimum: 1 + 2 + 1 = 4 critical points.

Connections with other topics

De Rham cohomology links smooth geometry, algebraic topology, and modern physics.

  • Algebraic topology — Related topic
  • Topological data analysis — Related topic
  • String theory — Related topic
  • Quantum physics — Related topic

Итоги

  • H^k_dR(M) = closed k-forms / exact k-forms; Betti numbers beta_k = dim H^k measure k-dimensional holes
  • De Rham theorem (1931): H^k_dR(M) cong H^k(M; R) - smooth data equals combinatorial data on topology
  • Hodge theorem (1941): every class in H^k has a unique harmonic representative (Delta omega = 0)
  • Hodge decomposition: Omega^k = H^k + im(d) + im(d*) - an orthogonal sum of three components
  • Kunneth formula and Mayer-Vietoris sequence: computational tools for cohomology of products and unions

Вопросы для размышления

  • De Rham's theorem: smooth forms carry the same topological information as combinatorial chains. Why is this a deep result, given that forms use the full smooth structure?
  • The Hodge decomposition singles out a 'best representative' in each cohomology class. Is there an analog in linear algebra or optimization?
  • Betti numbers are robust to deformations of the space. What does persistent homology do with this robustness for noisy data?

Связанные уроки

  • calc-28-manifold-int — Integration of forms over a manifold is the foundation of the de Rham isomorphism
  • calc-27-diff-forms — d^2 = 0 generates the cohomology groups
  • calc-26-divergence-theorem — Flux through a closed surface is the prototype of cohomological computations
De Rham Cohomology

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