Dyer–Lashof operations on Tate cohomology of finite groups

Langer, Martin

doi:10.2140/agt.2012.12.829

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Dyer–Lashof operations on Tate cohomology of finite groups

Martin Langer

Algebraic & Geometric Topology 12 (2012) 829–865

DOI: 10.2140/agt.2012.12.829

Abstract

Let $k = F_{p}$ be the field with $p > 0$ elements, and let $G$ be a finite group. By exhibiting an $E_{\infty}$ –operad action on $Hom (P, k)$ for a complete projective resolution $P$ of the trivial $k G$ –module $k$ , we obtain power operations of Dyer–Lashof type on Tate cohomology $Ĥ^{*} (G; k)$ . Our operations agree with the usual Steenrod operations on ordinary cohomology $H^{*} (G)$ . 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.

Keywords

Tate cohomology, Dyer–Lashof, cohomology operation, finite group

Mathematical Subject Classification 2010

Primary: 20J06, 55S12

References

Publication

Received: 18 June 2011

Revised: 22 November 2011

Accepted: 6 January 2012

Published: 17 April 2012

Authors

Martin Langer
Mathematisches Institut Rheinische Friedrich-Wilhelms-Universität Bonn Endenicher Allee 60 D-53115 Bonn Germany
http://www.math.uni-bonn.de/people/mlanger/

Volume 12, issue 2 (2012)