Volume 6, issue 1 (2006)

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Completed representation ring spectra of nilpotent groups

Tyler Lawson

Algebraic & Geometric Topology 6 (2006) 253–285

arXiv: 0902.4867


In this paper, we examine the “derived completion” of the representation ring of a pro-p group Gp 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 , 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[Gp] 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.

S-algebra, R-module, completion, Bousfield localization, representation ring
Mathematical Subject Classification 2000
Primary: 55P60
Secondary: 55P43, 19A22
Forward citations
Received: 11 April 2005
Revised: 31 October 2005
Accepted: 5 January 2006
Published: 26 February 2006
Tyler Lawson
Department of Mathematics
Massachusetts Institute of Technology
Cambridge MA 02139