Productive elements in group cohomology

Abstract

Let G be a finite group and k be a field of characteristic p > 0. A cohomology class ζ ∈ H n (G, k) is called productive if it annihilates Ext * kG (L ζ , L ζ ). 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. IntroductionLet G be a finite group and k be a field of characteristic p > 0. Let ζ ∈ H n (G, k) denote a nonzero cohomology class of degree n, where n 1. Associated to ζ, there is a unique kG-module homomorphismζ : Ω n k → k and the kG-module L ζ is defined as the kernel of this homomorphism. A cohomology class ζ is called productive if it annihilates the cohomology ring Ext * kG (L ζ , L ζ ). In this paper, we study the conditions for a cohomology class to be productive. Under the usual identification of H n (G, k) with the group U n (k, k) of n-fold kGmodule extensions of k by k, the cohomology class ζ is the extension class of an extension of the form

Dyer–Lashof operations on Tate cohomology of finite groups

“…The "only if" part of this theorem has been conjectured and independently proven in the case of ordinary cohomology classes by Yalçin [17], using connections to the existence of diagonal approximations of certain chain complexes.…”