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Dyer–Lashof operations on Tate cohomology of finite groups

Martin Langer

Algebraic & Geometric Topology 12 (2012) 829–865

Let k = Fp be the field with p > 0 elements, and let G be a finite group. By exhibiting an E–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 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.

Tate cohomology, Dyer–Lashof, cohomology operation, finite group
Mathematical Subject Classification 2010
Primary: 20J06, 55S12
Received: 18 June 2011
Revised: 22 November 2011
Accepted: 6 January 2012
Published: 17 April 2012
Martin Langer
Mathematisches Institut
Rheinische Friedrich-Wilhelms-Universität Bonn
Endenicher Allee 60
D-53115 Bonn