Rational local systems and connected finite loop spaces

TL;DR

This paper demonstrates that the category of rational local systems on connected finite loop spaces admits a simple algebraic model, extending known results for compact Lie groups and exploring the role of torsion and completion.

Contribution

It establishes an algebraic model for rational local systems on connected finite loop spaces, generalizing results for compact Lie groups and analyzing torsion and completion effects.

Findings

Rational local systems on connected finite loop spaces have algebraic models.

Special case recovered for rational cofree G-spectra when G is a compact Lie group.

Extension of algebraic modeling to categories associated with subgroups with connected Weyl groups.

Abstract

Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree $G$-spectra. More generally, we show that if $K$ is a closed subgroup of a compact Lie group $G$ such that the Weyl group $W_GK$ is connected, then a certain category of rational $G$-spectra `at $K$' has an algebraic model. For example, when $K$ is the trivial group, this is just the category of rational cofree $G$-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.