Dyer–Lashof operations on Tate cohomology of finite groups

Abstract

Let $k={\mathbb{F}}_p$ be the field with $p>0$ elements, and let $G$ be a finite group. By exhibiting an $E_{\infty }$–operad action on $Hom\left(P,k\right)$ for a complete projective resolution $P$ of the trivial $kG$–module $k$, we obtain power operations of Dyer–Lashof type on Tate cohomology ${Ĥ}^{\ast }\left(G;k\right)$. Our operations agree with the usual Steenrod operations on ordinary cohomology $H^{\ast }\left(G\right)$. We show that they are compatible (in a suitable sense) with products of groups, and (in certain cases) with the Evens norm map. These theorems provide tools for explicit computations of the operations for small groups $G$. We also show that the operations in negative degree are nontrivial.

As an application, we prove that at the prime $2$ these operations can be used to determine whether a Tate cohomology class is productive (in the sense of Carlson) or not.