The rational homotopy groups of virtual spheres for rank 1 compact Lie groups

TL;DR

This paper computes the rational stable stems for rank 1 compact Lie groups, demonstrating the effectiveness of algebraic models and highlighting limitations of tom Dieck splitting for desuspensions.

Contribution

It provides explicit calculations of rational representation-ring-graded stable stems for rank 1 groups, extending previous finite group results to Lie groups.

Findings

Effective algebraic models for G-spectra categories

Explicit rational stable stems for rank 1 groups

Limitations of tom Dieck splitting for desuspensions

Abstract

We calculate the rational representation-ring-graded stable stems for rank 1 groups, SU(2), SO(3), Pin (2), O(2), Spin(2) and SO(2), in the same spirit as the calculations for finite groups in arXiv:2205.02382 with J.D.Quigley. This illustrates the effectiveness of the algebraic models for these categories of G-spectra, and the way tom Dieck splitting fails for desuspensions. [v4: typos and tweaks in wording]