Geodesic
Dyer–Lashof operations on Tate cohomology of finite groups
Langer, Martin1
1Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany
Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 829-865
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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.

DOI : 10.2140/agt.2012.12.829
Classification : 20J06, 55S12
Keywords: Tate cohomology, Dyer–Lashof, cohomology operation, finite group

Langer, Martin  1

1 Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany 
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Langer, Martin. Dyer–Lashof operations on Tate cohomology of finite groups. Algebraic and Geometric Topology, Tome 12 (2012) no. 2, pp. 829-865. doi: 10.2140/agt.2012.12.829

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