Dyer-Lashof operations on Tate cohomology of finite groups

TL;DR

This paper introduces Dyer-Lashof power operations on Tate cohomology of finite groups using an E-infinity-operad action, extending the structure of cohomology and enabling explicit computations and applications at prime 2.

Contribution

It constructs Dyer-Lashof operations on Tate cohomology via an E-infinity-operad action, linking them to Steenrod operations and providing tools for computations and applications.

Findings

Operations agree with Steenrod on ordinary cohomology

Operations are compatible with group products and Evens norm map

Operations in negative degrees are non-trivial

Abstract

Let k be the field with p>0 elements, and let G be a finite group. By exhibiting an E-infinity-operad action on Hom(P,k) for a complete projective resolution P of the trivial kG-module k, we obtain power operations of Dyer-Lashof type on Tate cohomology H*(G; k). Our operations agree with the usual Steenrod operations on ordinary cohomology. 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 non-trivial. 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.