Cohomology, Bocksteins, and resonance varieties in characteristic 2

Abstract

We use the action of the Bockstein homomorphism on the cohomology ring H^{{\kern -0.25pt\raise 1.2pt\hbox to 0.5em{\tiny {∙ }}\hspace {-0.5pt}}}(X,\mathbb {Z}_2) of a finite-type CW-complex X X in order to define the resonance varieties of X X in characteristic 2 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 Z 2 \mathbb {Z}_2 has strong implications on the nature of the resonance varieties.