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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Quadratic maps and Bockstein closed group extensions

Authors: Jonathan Pakianathan and Ergün Yalçin
Journal: Trans. Amer. Math. Soc. 359 (2007), 6079-6110
MSC (2000): Primary 20J05; Secondary 17B50, 15A63
Published electronically: May 7, 2007
MathSciNet review: 2336317
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Abstract: Let $ E$ be a central extension of the form $ 0 \to V \to G \to W \to 0$ where $ V$ and $ W$ are elementary abelian $ 2$-groups. Associated to $ E$ there is a quadratic map $ Q: W \to 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^2(W,V)$ and an ideal $ I(q)$ in $ H^2(G, \mathbb{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 $ Q_{\mathfrak{gl}_n}: \mathfrak{gl}_n (\mathbb{F}_2)\to \mathfrak{gl}_n (\mathbb{F}_2)$ given by $ 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 \to M \to \widetilde{G} \to W \to 0$ for some $ \mathbb{Z} /4[W]$-lattice $ M$. In this situation, one may write $ \beta (q)=Lq$ for a ``binding matrix'' $ L$ with entries in $ H^1(W, \mathbb{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.

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Additional Information

Jonathan Pakianathan
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

Ergün Yalçin
Affiliation: Department of Mathematics, Bilkent University, Ankara, 06800, Turkey

Keywords: Group extensions, quadratic maps, group cohomology, restricted Lie algebras.
Received by editor(s): December 2, 2005
Published electronically: May 7, 2007
Additional Notes: The second author was partially supported by a grant from the Turkish Academy of Sciences (TÜBA-GEBİP/2005-16).
Article copyright: © Copyright 2007 American Mathematical Society

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