Cohomology jump loci of 3-manifolds

Abstract

The cohomology jump loci of a space X are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for X. We explore here the geometry of these varieties and the delicate interplay between them in the context of closed, orientable 3-dimensional manifolds and link complements. The classical multivariable Alexander polynomial plays an important role in this analysis. As an application, we derive some consequences regarding the formality and the existence of finite-dimensional models for such 3-manifolds.

Formality properties of finitely generated groups and Lie algebras

“…The non-1-formality of these groups can also be detected by suitable higher-order Massey products. We refer to [74] for a complete answer to the formality question for complements of 2-component links.…”

Poincaré duality and resonance varieties

“…In Appendix A we list the irreducible 3-forms µ A of rank at most 8, according to the classification from [22,13], together with the corresponding resonance varieties R 1 k (A). This work is pursued in [34], where we provide further applications to the study of cohomology jump loci of 3-manifolds.…”

Sigma-invariants and tropical varieties

“…We give in this section a brief review of the theory of characteristic varieties and Alexander polynomials. For details and further references we refer to [21,47,48,55,57] for the former, and [20,25,34,40,62] for the latter.…”

“…for all i ≤ q, where R i (X) ⊆ H 1 (X, C) are the resonance varieties associated to the cohomology algebra H * (X, C); see [56,57] for more on this. Letting R i (X, R) = R i (X) ∩ H 1 (X, R) be the real resonance varieties, the following inclusion then holds in the formal setting,…”

“…We show next how to recover this upper bound from our tropical bound; for the sake of brevity, we restrict to the case b 1 (M) ≥ 2. For more on the characteristic varieties of 3-manifolds, as well as their (exponential) tangent cones, we refer to [57]. Proof.…”