TY - CHAP

T1 - An introduction to algebraic models for rational G-spectra

AU - Barnes, David

AU - Kedziorek, Magdalena

PY - 2021/10/29

Y1 - 2021/10/29

N2 - 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 chapter 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-spectra in terms of a symmetric monoidal algebraic category. Having these two parts in the same place allows one to see clearly the analogy between the algebraic and topological classifications.

AB - 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 chapter 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-spectra in terms of a symmetric monoidal algebraic category. Having these two parts in the same place allows one to see clearly the analogy between the algebraic and topological classifications.

U2 - 10.1017/9781108942874.006

DO - 10.1017/9781108942874.006

M3 - Chapter (peer-reviewed)

SN - 9781108931946

VL - 474

T3 - London Mathematical Society Lecture Note Series

SP - 119

EP - 179

BT - Equivariant topology and derived algebra

A2 - Balchin, Scott

A2 - Barnes, David

A2 - Kędziorek, Magdalena

A2 - Szymik, Markus

PB - Cambridge University Press

ER -