PUB - Publikationen an der Universität Bielefeld

We define and study differential complexes of Alexander-Spanier type on metric measure spaces associated with (generally) unbounded non-local operators, such as operators of fractional Laplacian type. We show that these complexes can be used to approximate complexes of differential forms in a non-local-to-local convergence on the level of cores. Under an absolute continuity condition, we construct Hilbert complexes, observe invariance properties, and obtain associated self-adjoint Hodge Laplacians. For the case of _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 Poincaré lemma, and show that, in the compact Riemannian manifold case, the de Rham cohomology is recovered.