  We explore the relationship between a certain "abelian duality" property of
spaces and the propagation properties of their cohomology jump loci. To that
end, we develop the analogy between abelian duality spaces and those spaces
which possess what we call the "EPY property." The same underlying homological
algebra allows us to deduce the propagation of jump loci: in the former case,
characteristic varieties propagate, and in the latter, the resonance varieties.
We apply the general theory to arrangements of linear and elliptic hyperplanes,
as well as toric complexes, right-angled Artin groups, and Bestvina-Brady
groups. Our approach brings to the fore the relevance of the Cohen-Macaulay
condition in this combinatorial context.
