Completed representation ring spectra of nilpotent groups

Details Completed representation ring spectra of nilpotent groups Completed representation ring spectra of nilpotent groups

2009-02-27

arxiv.org/abs/0902.4867v1

Algebr. Geom. Topol. 6 (2006) 253-285

Mathematics

Algebraic Topology

This is the version published by Algebraic & Geometric Topology on 26 February 2006

Scientific paper

10.2140/agt.2006.6.253

In this paper, we examine the `derived completion' of the representation ring of a pro-p group G_p^ with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg-MacLane spectrum HZ, and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor R[-] from groups to ring spectra, and show that the map R[G_p^] --> R[G] becomes an equivalence after completion when G is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the p-adic Heisenberg group.

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