Topology-Aware DEC

This page documents the cohomology/harmonic layer on closed orientable triangulated surfaces.

Mathematical object

For edge 1-cochains ω ∈ Ω¹, we use the discrete decomposition

ω = dα + δβ + h

with:

dα exact (im(d0)),
δβ coexact (im(δ2)),
h harmonic (ker(d1) ∩ ker(δ1)).

The harmonic dimension is the first Betti number β1 (for closed genus-g surfaces, β1 = 2g).

Cochain space / degree

β-functions and cycle/cohomology basis: topology of the primal mesh.
harmonic_basis, project_*, and hodge_decomposition_full: edge 1-cochains (length = nE).
Potentials:
- α on vertices (Ω⁰, length nV),
- β on faces (Ω², length nF).

Orientation convention

Edge orientation is canonical (i,j) with i < j.
Signed cycle entries:
- +e means traversal along canonical edge orientation,
- -e means opposite traversal.
Inner product is the Hodge metric:
- ⟨a,b⟩ = a' * star1 * b.

Storage convention

All 1-forms are dense vectors in edge ordering from build_topology(mesh).edges.
Harmonic basis is an nE × β1 matrix; each column is one harmonic generator.

API

betti_numbers
first_betti_number
cycle_basis
cohomology_basis_1
harmonic_basis
project_harmonic
project_exact
project_coexact
hodge_decomposition_full
is_closed_form
is_coclosed_form
harmonic_residuals

Minimal example

using FrontIntrinsicOps
using LinearAlgebra

mesh = generate_torus(2.0, 0.7, 20, 24)
geom = compute_geometry(mesh)
dec = build_dec(mesh, geom)

b = betti_numbers(mesh)
H = harmonic_basis(mesh, geom, dec)

ω = dec.d0 * [p[3] for p in mesh.points] + 0.4 .* H[:, 1]
decomp = hodge_decomposition_full(ω, mesh, geom, dec; basis=H)

(β=b, nbasis=size(H, 2), residual=norm(decomp.residual))

Limitations and non-goals

Closed orientable surfaces only in v1; open-surface harmonic tools are not implemented here.
Basis vectors are deterministic by construction, but harmonic sign/basis choices remain topologically equivalent (not unique analytically).
Implementation is low-order simplicial DEC; no symbolic FEEC framework is introduced.