Bockstein Closed 2-Group Extensions and Cohomology of Quadratic Maps

A central extension of the form $E: 0 \to V \to G \to W \to 0$, where $V$ and $W$ are elementary abelian 2-groups, is called Bockstein closed if the components $q_i \in H^*(W, \FF_2)$ 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 $Q : W \to V$ associated to the extensions $E$ of the above form.