Cohomology, Bocksteins, and resonance varieties in characteristic 2

Abstract

We use the action of the Bockstein homomorphism on the cohomology ring $H^{\ast }(X,{\mathbb{Z}}_2)$ 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 Poincaré duality over ${\mathbb{Z}}_2$ has strong implications on the nature of the resonance varieties.

25 pages; accepted for publication in Contemporary Mathematics

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