# Cohomology jump loci of 3-manifolds

## TL;DR

This paper investigates the geometric properties of cohomology jump loci in 3-manifolds, examining their relationship with Alexander polynomials and implications for manifold formality and models.

## Contribution

It provides new insights into the geometry of characteristic and resonance varieties in 3-manifolds and links these to classical invariants like the Alexander polynomial.

## Key findings

- Analysis of the geometry of cohomology jump loci
- Connections between jump loci and Alexander polynomial
- Implications for formality and finite-dimensional models

## 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.