Title: Quadratic Maps and Bockstein Closed Group Extensions 

Authors: Jonathan Pakianathan and Erg\" un Yal\c c\i n

Mailing addresses: 

Jonathan Pakianathan
Dept. of Mathematics 
University of Rochester, 
Rochester, NY 14627 U.S.A. 

Ergun Yalcin
Bilkent Universitesi
Matematik Bolumu
Bilkent, Ankara, 
06800, Turkey 

Paper Status: to appear in Transactions of A.M.S.

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 $H^2(G, \ZZ /2)$
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 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 $Q_{\gl _n}: \gl _n (\FF _2)
\to \gl _n (\FF _2)$ given by $Q(\MA)= \MA +\MA ^2$ 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 $\ZZ /4[W]$-lattice $M$. In
this situation, one may write $\beta (q)=Lq$ for a ``binding
matrix'' $L$ with entries in $H^1(W, \ZZ /2)$. 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.