Bockstein closed 2-group extensions and cohomology of quadratic maps

A central extension of the form E:0→V→G→W→0, where V and W are elementary abelian 2-groups, is called Bockstein closed if the components qi∈H⁎(W,F2) of the extension class of E generate an ideal which is closed under the Bockstein operator. In this paper, we study the cohomology ring of G when E is a Bockstein closed 2-power exact extension. The mod-2 cohomology ring of G has a simple form and it is easy to calculate. The main result of the paper is the calculation of the Bocksteins of the generators of the mod-2 cohomology ring using an Eilenberg–Moore spectral sequence. We also find an interpretation of the second page of the Bockstein spectral sequence in terms of a new cohomology theory that we define for Bockstein closed quadratic maps Q:W→V associated to the extensions E of the above form.