A tensor product approach to non-local differential complexes

TL;DR

This paper develops a tensor product framework for non-local differential complexes on metric measure spaces, extending classical concepts like Hodge Laplacians and cohomology to fractional and non-local operators.

Contribution

It introduces a tensor product approach to non-local differential complexes, establishing invariance, self-adjointness, and cohomological properties for fractional Laplacian operators.

Findings

Defined Hilbert complexes for non-local operators

Proved a Mayer-Vietoris principle and Poincaré lemma

Reconstructed de Rham cohomology in the Riemannian case

Abstract

We study differential complexes of Kolmogorov-Alexander-Spanier type on metric measure spaces associated with unbounded non-local operators, such as operators of fractional Laplacian type. We define Hilbert complexes, observe invariance properties and obtain self-adjoint non-local analogues of Hodge Laplacians. For $d$-regular measures and operators of fractional Laplacian type we provide results on removable sets in terms of Hausdorff measures. We prove a Mayer-Vietoris principle and a Poincar\'e lemma and verify that in the compact Riemannian manifold case the deRham cohomology can be recovered.