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

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

2010-01-05

arxiv.org/abs/1001.0771v3

Mathematics

Algebraic Topology

Final version, to appear in Advances in Mathematics

Scientific paper

10.1016/j.aim.2011.03.014

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.

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