## Algebraic & Geometric Topology

### Completed representation ring spectra of nilpotent groups

Tyler Lawson

#### Abstract

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 $ℍℤ$, 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 $R[Gp∧]→R[G]$ 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 6, Number 1 (2006), 253-285.

Dates
Revised: 31 October 2005
Accepted: 5 January 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796512

Digital Object Identifier
doi:10.2140/agt.2006.6.253

Mathematical Reviews number (MathSciNet)
MR2199460

Zentralblatt MATH identifier
1118.55009

#### Citation

Lawson, Tyler. Completed representation ring spectra of nilpotent groups. Algebr. Geom. Topol. 6 (2006), no. 1, 253--285. doi:10.2140/agt.2006.6.253. https://projecteuclid.org/euclid.agt/1513796512

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