Quadratic maps and Bockstein closed group extensions
Transactions of the American Mathematical Society
American Mathematical Society
6079 - 6110
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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 H2(W, V) and an ideal I(q) in H2(G, ℤ/2) which is generated by the components of q. We say that E is Bockstein closed if I(q) is an ideal closed under the Bockstein operator. We find a direct condition on the quadratic map Q that characterizes when the extension is Bockstein closed. Using this characterization, we show for example that quadratic maps induced from the fundamental quadratic map Qgln: gln(F2) → gln(F2) given by Q(A) = A + A2 yield Bockstein closed extensions. On the other hand, 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 H1(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. © 2007 American Mathematical Society.