Bockstein Closed Central Extensions of Elementary Abelian 2-Groups I: Binding Operators

Let E be a central extension of the form 0 → V → G → W → 0 where V and W are elementary abelian 2-groups. Associated to E there is a quadratic map Q : W → V given by the 2-power map which uniquely determines the extension. This quadratic map also determines the extension class q of the extension in H(W,V ) and an ideal I(q) in H(G,Z/2) which is generated by the components of q. We say E is Bockstein closed if I(q) is an ideal closed under the Bockstein operator. We find a direct condition on the quadratic form Q that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic forms induced from the fundamental quadratic form Qgl n : gl n (F2) → gl n (F2) given by Q(A) = A + A 2 yield Bockstein closed extensions. It is well known that an extension is Bockstein closed if and only if it lifts to an extension 0 → M → G̃ → W → 0 for some Z/4[W ]-lattice M . In this situation, one may write β(q) = Lq for a “binding matrix” L with entries in H(W,Z/2). We find a direct way to calculate the module structure of M in terms of L. Using this, we study extensions where the lattice M is diagonalizable/triangulable and find interesting equivalent conditions to these properties. 2000 Mathematics Subject Classification. Primary: 20J05; Secondary: 17B50, 15A63.