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# Homology, Homotopy and Applications

## Volume 13 (2011)

### Number 1

### Productive elements in group cohomology

Pages: 381 – 401

DOI: http://dx.doi.org/10.4310/HHA.2011.v13.n1.a15

#### Author

#### Abstract

Let $G$ be a finite group and $k$ be a field of characteristic $p \gt 0$. A cohomology class $\zeta\in H^n (G,k)$ is called productive if it annihilates $\operatorname{Ext}^*_{kG}(L_\zeta,L_\zeta)$. We consider the chain complex $\mathbf{P}(\zeta)$ 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 $\Delta : \mathbf{P}(\zeta)\to \mathbf{P}(\zeta)\otimes \mathbf{P}(\zeta)$ such that $(\operatorname{id} \otimes \epsilon) \Delta \simeq \operatorname{id}$ and $(\epsilon \otimes \operatorname{id}) \Delta \simeq \operatorname{id}$. Using the Postnikov decomposition of $\mathbf{P}(\zeta) \otimes \mathbf{P}(\zeta)$, 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.

#### Keywords

group cohomology; chain complex; diagonal approximation

#### 2010 Mathematics Subject Classification

20C20, 20J06, 57S17