Productive elements in group cohomology

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 $\rm{Ext}^*_{kG}(L_\zeta,L_\zeta)$. We consider the chain complex $\mathbf{P}(\zeta)$ 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 $\Delta : \mathbf{P}(\zeta)\to \mathbf{P}(\zeta)\otimes \mathbf{P}(\zeta)$ such that $(\rm{id} \otimes \epsilon) \Delta \simeq \rm{id}$ and $(\epsilon \otimes \rm{id}) \Delta \simeq \rm{id}$. Using the Postnikov decomposition of $\mathbf{P}(\zeta) \otimes \mathbf{P}(\zeta)$, 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 Carlson and Langer on productive elements.