Title: Bockstein Closed 2-Group Extensions and 
Cohomology of Quadratic Maps


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


Status: To appear in Journal of Algebra


Address 1: 
Department of Mathematics, 

University of Rochester, 

Rochester, NY 14627 U.S.A. 

E-mail: jonpak@math.rochester.edu 


Address 2: 
Department of Mathematics, Bilkent University,
06800 Bilkent, Ankara, Turkey
Email: yalcine@fen.bilkent.edu.tr 


Abstract:
A central extension of the form $E: 0 \to V \to G \to W \to 0$,

where $V$ and $W$ are elementary abelian $2$-groups, is called

Bockstein closed if the components  $ q_i \in H^*(W, \FF _2 )$ 
of 
the extension class of $E$ generate an ideal which is closed 
under 
the Bockstein operator. In this paper, we study the 
cohomology ring 
of $G$ when $E$ is a Bockstein closed $2$-power 
exact extension. The 
mod-$2$ cohomology ring of $G$ has a simple 
form and it is easy to 
calculate. The main result of the paper is 
the calculation of the Bocksteins of the generators of the mod-$2$ 
cohomology ring using an 
Eilenberg-Moore spectral sequence. We also 
find an interpretation of 
the second page of the Bockstein spectral 
sequence in terms of a new
 cohomology theory that we define for 
Bockstein closed quadratic maps $Q : W \to V$ associated to the 
extensions $E$ of the above form.