Title: Completion of G-spectra and stable maps between classifying spaces

Author: Kári Ragnarsson

Status: Submitted

Abstract: We prove structural theorems  for computing the completion of 
a G-spectrum at the augmentation ideal of the Burnside ring of a finite 
group G. First we show that a G-spectrum can be replaced by a spectrum 
obtained by allowing only isotropy groups of prime power order without 
changing the homotopy type of the completion. We then show that this 
completion can be computed as a homotopy colimit of completions of spectra 
obtained by further restricting isotropy to one prime at a time, and that 
these completions can be computed in terms of completion at a prime.
 
As an application, we show that the spectrum of stable maps from BG to the 
classifying space of a compact Lie group K splits non-equivariantly as a 
wedge sum of p-completed suspension spectra of classifying spaces of certain 
subquotients of the product of G by K. In particular this describes the 
dual of BG.