Title: Productive elements in group cohomology


Author: Erg\"un Yal\c c\i n

Address: 
Department of Mathematics, 
Bilkent University,
06800 Bilkent, Ankara, 
Turkey

Email: yalcine@fen.bilkent.edu.tr 

Status: Appeared in Homology, Homotopy and Applications.

Abstract: Let $G$ be a finite group and $k$ be a field of characteristic $p>0$. 
A cohomology class $\zeta \in H^n(G,k)$ is called productive if it annihilates 
$\Ext^*_{kG}(L_{\zeta} ,L_{\zeta})$. We consider the chain complex $\bPz$ of projective 
$kG$-modules which has the  homology of an $(n-1)$-sphere and whose $k$-invariant 
is $\zeta$ under a certain polarization. We show that $\zeta$ is productive if and only 
if there is a chain map $\Delta: \bPz \to \bPz \otimes \bPz$ such that 
$(\id \otimes \epsilon)\Delta\simeq \id$ and $(\epsilon \otimes \id)\Delta \simeq \id$. 
Using the Postnikov decomposition of $\bPz \otimes \bPz$, we prove that there is a unique 
obstruction for constructing a chain map $\Delta$ satisfying these properties. Studying this
obstruction more closely, we obtain theorems of J.F. Carlson and M. Langer on productive elements.