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

J. P. C. Greenlees

TL;DR

The paper determines the rational representation-ring-graded stable stems [S^V,S^W]^G_* for all rank-1 compact Lie groups by employing algebraic models of rational G-spectra and a block-decomposition into cyclic, dihedral, and isolated components. The approach reduces to fixed-point representation theory, with the circle group SO(2) providing the key input; Adams spectral sequences collapse to short exact sequences, allowing explicit dimension formulas via the fixed-point data d_U(s) and d'_U(t). The main results describe the cyclic and dihedral contributions and isolate fixed-point data for the remaining blocks across SO(2), O(2), SO(3), Pin(2), SU(2), and Spin(2), yielding complete descriptions of [S^0,S^U]^H_* for each rank-1 group. This work demonstrates the effectiveness of the rational-algebraic models in organizing G-spectra computations, clarifies how desuspensions interact with blocks, and generalizes finite-group calculations to rank-1 settings with explicit representation-theoretic control.

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]

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

TL;DR

The paper determines the rational representation-ring-graded stable stems [S^V,S^W]^G_* for all rank-1 compact Lie groups by employing algebraic models of rational G-spectra and a block-decomposition into cyclic, dihedral, and isolated components. The approach reduces to fixed-point representation theory, with the circle group SO(2) providing the key input; Adams spectral sequences collapse to short exact sequences, allowing explicit dimension formulas via the fixed-point data d_U(s) and d'_U(t). The main results describe the cyclic and dihedral contributions and isolate fixed-point data for the remaining blocks across SO(2), O(2), SO(3), Pin(2), SU(2), and Spin(2), yielding complete descriptions of [S^0,S^U]^H_* for each rank-1 group. This work demonstrates the effectiveness of the rational-algebraic models in organizing G-spectra computations, clarifies how desuspensions interact with blocks, and generalizes finite-group calculations to rank-1 settings with explicit representation-theoretic control.

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]

Paper Structure

This paper contains 37 sections, 8 theorems, 55 equations.

Table of Contents

  1. Context
  2. Precursors
  3. The groups
  4. Conventions
  5. Contents
  6. The group $SO(2)$
  7. Subgroups
  8. Representations
  9. Notation
  10. The standard model
  11. The calculation
  12. The group $O(2)$
  13. Subgroups in blocks
  14. Models
  15. Representations
  16. ...and 22 more sections

Key Result

Lemma 2.1

The vector space $(\Sigma^{U}\mathbb{I})_0$ is finite dimensional: it is a sum with one copy of $\mathbb$Q$$ for each $s$ with $d_U(s)<0$. The connecting map $\mathbb$Q$\longrightarrow \Sigma^{U}\mathbb{I}$ is the diagonal. In particular it is zero if $d_U(s)\geq 0$ for all $s$ and it is nonzero if

Theorems & Definitions (8)

  • Lemma 2.1
  • Corollary 2.2
  • Lemma 4.1
  • Corollary 4.2
  • Corollary 4.3
  • Lemma 5.1
  • Corollary 5.2
  • Lemma 6.1