Cohomology, Bocksteins, and resonance varieties in characteristic 2

TL;DR

This paper develops a framework for defining and analyzing resonance varieties of finite CW-complexes in characteristic 2 using Bockstein homomorphisms, with applications to manifolds and Poincaré duality.

Contribution

It introduces a novel approach to resonance varieties in characteristic 2 via Bockstein actions and extends the theory to differential graded algebras, with illustrative examples.

Findings

Resonance varieties are characterized using Bockstein homomorphisms in characteristic 2.

Poincaré duality influences the structure of resonance varieties in closed manifolds.

The framework applies to finite-type CW-complexes and differential graded algebras.

Abstract

We use the action of the Bockstein homomorphism on the cohomology ring $H^*(X,\mathbb{Z}_2)$ of a finite-type CW-complex $X$ in order to define the resonance varieties of $X$ in characteristic 2. Much of the theory is done in the more general framework of the Maurer-Cartan sets and the resonance varieties attached to a finite-type commutative differential graded algebra. We illustrate these concepts with examples mainly drawn from closed manifolds, where Poincar\'e duality over $\mathbb{Z}_2$ has strong implications on the nature of the resonance varieties.