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The cohomology jump loci of a space $X$ are of two basic types: thecharacteristic varieties, defined in terms of homology with coefficients inrank one local systems, and the resonance varieties, constructed frominformation encoded in either the cohomology ring, or an algebraic model for$X$. We explore here the geometry of these varieties and the delicate interplaybetween them in the context of closed, orientable 3-dimensional manifolds andlink complements. The classical multivariable Alexander polynomial plays animportant role in this analysis. As an application, we derive some consequencesregarding the formality and the existence of finite-dimensional models for such3-manifolds.

Cohomology jump loci of 3-manifolds