Completion of $G$-spectra and stable maps between classifying spaces

TL;DR

This paper develops new structural theorems for computing G-spectrum completions at the Burnside ring's augmentation ideal, simplifying the process via isotropy restrictions and prime localizations, with applications to stable maps between classifying spaces.

Contribution

It introduces a method to compute G-spectrum completions by restricting isotropy groups and prime localizations, and applies this to analyze stable maps between classifying spaces.

Findings

G-spectrum completion can be simplified by isotropy restrictions.

Completion at a prime can be computed via homotopy colimits of prime-specific completions.

Stable maps from BG to classifying spaces split into wedge sums of p-completed spectra.

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 and K. In particular this describes the dual of BG.