Quadratic maps and Bockstein closed group extensions

Abstract

Let E E be a central extension of the form 0 → V → G → W → 0 0 \to V \to G \to W \to 0 where V V and W W are elementary abelian 2 2 -groups. Associated to E E there is a quadratic map Q : W → V Q: W \to V , given by the 2 2 -power map, which uniquely determines the extension. This quadratic map also determines the extension class q q of the extension in H 2 ( W , V ) H^2(W,V) and an ideal I ( q ) I(q) in H 2 ( G , Z / 2 ) H^2(G, \mathbb {Z} /2) which is generated by the components of q q . We say that E E is Bockstein closed if I ( q ) I(q) is an ideal closed under the Bockstein operator. We find a direct condition on the quadratic map Q 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 Q g l n : g l n ( F 2 ) → g l n ( F 2 ) Q_{\mathfrak {gl}_n}: \mathfrak {gl}_n (\mathbb {F}_2)\to \mathfrak {gl}_n (\mathbb {F}_2) given by Q ( A ) = A + A 2 Q(\mathbb {A})= \mathbb {A} +\mathbb {A} ^2 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 → G ~ → W → 0 0 \to M \to \widetilde {G} \to W \to 0 for some Z / 4 [ W ] \mathbb {Z} /4[W] -lattice M M . In this situation, one may write β ( q ) = L q \beta (q)=Lq for a “binding matrix” L L with entries in H 1 ( W , Z / 2 ) H^1(W, \mathbb {Z}/2) . We find a direct way to calculate the module structure of M M in terms of L L . Using this, we study extensions where the lattice M M is diagonalizable/triangulable and find interesting equivalent conditions to these properties.