Let G be a finite group and k be a field of characteristic p > 0. A cohomology class z ∈ Hn(G,k) is called productive if it annihilates Ext*kG(Lz ,Lz). We consider the chain complex P(z) of projective kG-modules which has the homology of an (n-1)-sphere and whose k-invariant is z under a certain polarization. We show that z is productive if and only if there is a chain map Δ: P(z) → P(z) ⊗P(z) such that (id ⊗e)Δ ≈ id and (e⊗id)Δ ≈ id. Using the Postnikov decomposition of P(z) ⊗P(z), we prove that there is a unique obstruction for constructing a chain map Δ satisfying these properties. Studying this obstruction more closely, we obtain theorems of Carlson and Langer on productive elements.
Homology, Homotopy and Applications, Vol. 13 (2011), No. 1, pp.381-401.
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