PRODUCTIVE ELEMENTS IN GROUP COHOMOLOGY

Let G be a finite group and k be a field of characteristic p > 0. A cohomology class zeta is an element of H-n (G, k) is called productive if it annihilates Ext(kG)*(L-zeta, L-zeta). We consider the chain complex P(zeta) of projective k G-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: P(zeta) -> P(zeta) circle times P(zeta) such that (id circle times epsilon) Delta similar or equal to id and (epsilon circle times id) Delta similar or equal to id. Using the Postnikov decomposition of P(zeta) circle times 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.