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,F 2) 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. © 2012 Elsevier Inc.