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Spaces of subgroups of toral groups

J. P. C. Greenlees

TL;DR

This work develops a systematic, cohomology-driven framework to classify conjugacy classes of subgroups of toral extensions $G$ of the form $1\to \mathbb T\to G\to W\to 1$, and to describe the topology and normalizer data on the space $\mathfrak X_G = \mathrm Sub}(G)/G$. Central to the method is Pontrjagin duality, which translates subgroups of the torus into $W$-invariant sublattices of the dual lattice, together with a ramification sequence that ties splittings, extensions, and conjugacy classes to $H^i(W; -)$ for carefully chosen modules $\Lambda_S$. The paper provides explicit rank-2 toral cases, detailing the cohomology computations, lattice normalizers, Weyl groups, and component structures, and it demonstrates how these invariants assemble into the Balmer spectrum for rational $G$-spectra. The results offer concrete, rank-2 illustrations of how algebraic models for rational equivariant cohomology theories can be built from subgroup data and lattice duality, guiding future work on more complex groups such as $SU(3)$ and beyond. The overall contribution is a robust, computable blueprint for translating subgroup-theoretic data into algebraic models for equivariant rational homotopy theory.

Abstract

We study the space of conjugacy classes of subgroups of a compact Lie group G whose identity component is a torus, and consider how various invariants of subgroups behave as sheaves over this space. This feeds in to the author's programme to give algebraic models of rational G-equivariant cohomology theories. The methods are illustrated by making the outcome explicit for all toral subgroups of compact connected rank 2 groups.

Spaces of subgroups of toral groups

TL;DR

This work develops a systematic, cohomology-driven framework to classify conjugacy classes of subgroups of toral extensions of the form , and to describe the topology and normalizer data on the space . Central to the method is Pontrjagin duality, which translates subgroups of the torus into -invariant sublattices of the dual lattice, together with a ramification sequence that ties splittings, extensions, and conjugacy classes to for carefully chosen modules . The paper provides explicit rank-2 toral cases, detailing the cohomology computations, lattice normalizers, Weyl groups, and component structures, and it demonstrates how these invariants assemble into the Balmer spectrum for rational -spectra. The results offer concrete, rank-2 illustrations of how algebraic models for rational equivariant cohomology theories can be built from subgroup data and lattice duality, guiding future work on more complex groups such as and beyond. The overall contribution is a robust, computable blueprint for translating subgroup-theoretic data into algebraic models for equivariant rational homotopy theory.

Abstract

We study the space of conjugacy classes of subgroups of a compact Lie group G whose identity component is a torus, and consider how various invariants of subgroups behave as sheaves over this space. This feeds in to the author's programme to give algebraic models of rational G-equivariant cohomology theories. The methods are illustrated by making the outcome explicit for all toral subgroups of compact connected rank 2 groups.
Paper Structure (94 sections, 36 theorems, 106 equations)

This paper contains 94 sections, 36 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Contents
  3. Motivation
  4. Organization
  5. Associated work in preparation
  6. Subgroups of toral groups (generalities)
  7. Pontrjagin duality for subgroups of toral groups
  8. Crude partition
  9. Subgroups of a torus
  10. A little more duality
  11. Classifying subgroups of toral groups
  12. Splittings
  13. Subgroups up to conjugacy
  14. The ramification sequence
  15. Variation of conjugacy classes
  16. ...and 79 more sections

Key Result

Lemma 2.1

For a toral group $G$ as above, the space $\mathfrak{X}_G=\mathrm{Sub}(G)/G$ of conjugacy classes of subgroups of the toral group $G$ is partitioned into pieces, $\mathcal{}$V$^G_{ \overline$H$}$ one for each conjugacy class of subgroups $\overline$H$$ of $W$. If $\overline$H$\subseteq W$, the set is clopen in the Hausdorff metric topology and closed under passage to cotoral subgroups. Furthermor

Theorems & Definitions (42)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • Example 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Example 3.4
  • Corollary 3.5
  • Remark 3.6
  • Lemma 4.1
  • ...and 32 more