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2012 Dyer–Lashof operations on Tate cohomology of finite groups
Martin Langer
Algebr. Geom. Topol. 12(2): 829-865 (2012). DOI: 10.2140/agt.2012.12.829


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.


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Martin Langer. "Dyer–Lashof operations on Tate cohomology of finite groups." Algebr. Geom. Topol. 12 (2) 829 - 865, 2012.


Received: 18 June 2011; Revised: 22 November 2011; Accepted: 6 January 2012; Published: 2012
First available in Project Euclid: 19 December 2017

zbMATH: 1278.20071
MathSciNet: MR2914620
Digital Object Identifier: 10.2140/agt.2012.12.829

Primary: 20J06 , 55S12

Keywords: cohomology operation , Dyer–Lashof , Finite group , Tate cohomology

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.12 • No. 2 • 2012
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