#### Volume 19, issue 7 (2019)

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An algebraic model for rational toral $G$–spectra

### David Barnes, John Greenlees and Magdalena Kędziorek

Algebraic & Geometric Topology 19 (2019) 3541–3599
##### Abstract

For $G$ a compact Lie group, toral $G$–spectra are those rational $G$–spectra whose geometric isotropy consists of subgroups of a maximal torus of $G\phantom{\rule{0.3em}{0ex}}$. The homotopy category of rational toral $G$–spectra is a retract of the category of all rational $G$–spectra.

We show that the abelian category of Greenlees (Algebr. Geom. Topol. 16 (2016) 1953–2019) gives an algebraic model for the toral part of rational $G$–spectra. This is a major step in establishing an algebraic model for all rational $G$–spectra for any compact Lie group $G\phantom{\rule{0.3em}{0ex}}$.

##### Keywords
equivariant cohomology, rational equivariant spectra, algebraic models, model category
##### Mathematical Subject Classification 2010
Primary: 55N91
Secondary: 55P42, 55P60