[Paper Review] An introduction to algebraic models for rational G-spectra

David Barnes, Magdalena Kędziorek|arXiv (Cornell University)|Apr 3, 2020

Homotopy and Cohomology in Algebraic Topology45 references4 citations

TL;DR

This paper provides a comprehensive introduction to algebraic models for rational G-spectra, establishing a Quillen equivalence between the category of rational G-spectra and an algebraic category of differential objects in a graded abelian category. It unifies the classification of rational Mackey functors and rational G-spectra via Burnside ring actions, change of groups functors, and cellularisation, with key results for finite groups, SO(3), O(2), and toral G-spectra using formality and algebraicisation techniques.

ABSTRACT

The project of Greenlees et al. on understanding rational G-spectra in terms of algebraic categories has had many successes, classifying rational G-spectra for finite groups, SO(2), O(2), SO(3), free and cofree G-spectra as well as rational toral G-spectra for arbitrary compact Lie groups. This paper provides an introduction to the subject in two parts. The first discusses rational G-Mackey functors, the action of the Burnside ring and change of group functors. It gives a complete proof of the well-known classification of rational Mackey functors for finite G. The second part discusses the methods and tools from equivariant stable homotopy theory needed to obtain algebraic models for rational G-spectra. It gives a summary of the key steps in the classification of rational G-spectrain terms of a symmetric monoidal algebraic category. Having these two parts in the same place allows one to clearly see the analogy between the algebraic and topological classifications.

Motivation & Objective

• To provide a self-contained introduction to algebraic models for rational G-spectra, bridging equivariant homotopy theory and algebraic categories.
• To clarify the structural analogy between the classification of rational Mackey functors and rational G-spectra via shared tools like Burnside ring actions and change of groups functors.
• To demonstrate how the classification of rational Mackey functors serves as a template for classifying rational G-spectra across various compact Lie groups.
• To extend the framework to toral G-spectra using cellularisation and formality arguments, particularly for SO(3) and O(2).
• To support the construction of monoidal algebraic models by analyzing idempotents in the rational Burnside ring and their role in localisations and splittings.

Proposed method

• Classify rational Mackey functors for finite G using the action of the rational Burnside ring and decomposition via Weyl group group rings.
• Use change of group functors (restriction, induction, coalescence) to relate Mackey functors across subgroups and conjugacy classes.
• Apply the Cellularisation Principle and Bousfield localisations to decompose rational G-spectra into algebraic components via idempotents in the rational Burnside ring.
• Construct algebraic models via algebraicisation of ring spectra (e.g., E_H) using Shipley’s work, reducing to modules over differential graded algebras.
• Leverage formality of commutative dgas to simplify algebraic models, removing cellularisation steps in key cases like SO(2) and toral G-spectra.
• Use Quillen equivalences to relate the derived category of rational G-spectra to differential objects in an abelian category A(G), establishing algebraic models.

Experimental results

Research questions

• RQ1 How can rational Mackey functors for a finite group G be classified using the rational Burnside ring and Weyl group group rings?
• RQ2 What is the role of change of group functors in relating rational Mackey functors across subgroups and conjugacy classes?
• RQ3 How do idempotents in the rational Burnside ring enable the splitting of rational G-spectra into algebraic components?
• RQ4 In what way does the Cellularisation Principle allow the reduction of rational toral G-spectra to cellularisations of rational toral N-spectra for N the normaliser of a maximal torus?
• RQ5 How do formality arguments and algebraicisation techniques simplify the construction of algebraic models for rational G-spectra?

Key findings

• The category of rational Mackey functors for a finite group G is equivalent to a product of modules over group rings of Weyl groups of conjugacy classes of subgroups.
• The classification of rational Mackey functors is achieved via a diagonal decomposition and Morita equivalence, with the rational Burnside ring acting as the ring of endomorphisms.
• For compact Lie groups, rational G-spectra are Quillen equivalent to differential objects in an abelian category A(G), with A(G) of injective dimension equal to the rank of G.
• The classification of rational toral G-spectra is reduced to a cellularisation of rational toral N-spectra, where N is the normaliser of a maximal torus, via a Quillen equivalence.
• Formality of the commutative differential graded algebras arising from algebraicisation allows removal of cellularisation, yielding a clean algebraic model for rational toral G-spectra.
• The algebraic model for rational G-spectra is compatible with monoidal structures in key cases, such as SO(3) and O(2), enabling applications to commutative ring spectra.