Productive elements in group cohomologyi

Abstract

Let G be a finite group and k be a field of characteristic p > 0. A cohomology class ζ ∈ Hn(G, k) is called productive if it annihilates Ext∗kG(Lζ , Lζ ). We consider the chain complex P(ζ) of projective kG-modules which has the homology of an (n − 1)-sphere and whose k-invariant is ζ under a certain polarization. We show that ζ is productive if and only if there is a chain map ∆: P(ζ) → P(ζ) ⊗ P(ζ) such that (id ⊗ )∆ 'id and ( ⊗ id)∆ ' id. Using the Postnikov decomposition of P(ζ) ⊗ P(ζ), we prove that there is a unique obstruction for constructing a chain map ∆ satisfying these properties. Studying this obstruction more closely, we obtain theorems of Carlson and Langer on productive elements.