  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.
