Extend Hodge-theoretic identification of cohomology to diffusion geometry beyond manifolds

Establish whether the identification of de Rham cohomology groups with the kernel of the Hodge Laplacian (the Hodge theorem) extends from smooth manifolds to general spaces equipped with diffusion geometry (e.g., Markov triples on measure spaces). Specifically, prove that harmonic differential forms computed via the diffusion-geometry Hodge Laplacian represent cohomology classes in non-manifold settings, providing a formal guarantee of H^k(M,R) ≅ ker(Δ^(k)) beyond the manifold case.

Background

The paper uses the Hodge Laplacian Δ^k computed via diffusion geometry to infer de Rham cohomology classes from data. On smooth manifolds, Hodge theory guarantees that harmonic forms (ker Δ^k) represent cohomology classes, but the authors employ these tools on general, possibly non-manifold data.

They state there is no formal guarantee that this correspondence holds beyond manifolds and explicitly conjecture that such a connection exists. Formalizing this would justify using harmonic forms from diffusion geometry as topological descriptors on general data geometries.