Productive elements in group cohomology

TL;DR

This paper characterizes productive elements in the cohomology of finite groups using chain complexes and Postnikov decompositions, providing new criteria and obstructions for their existence.

Contribution

It introduces a chain complex approach and identifies a unique obstruction to constructing chain maps that characterize productive elements in group cohomology.

Findings

A chain map criterion for productive elements.

Identification of a unique obstruction to chain map construction.

Connections to and generalizations of Carlson and Langer's theorems.

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 Carlson and Langer on productive elements.