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A Spectral Sequence for Equidimensional Actions of Compact Lie Groups

Paweł Raźny

TL;DR

The paper develops a Leray–Serre–type spectral sequence for equidimensional actions of a connected compact Lie group $G$ on a compact manifold $M$, bridging the basic cohomology of the orbit foliation $H^ullet(M/\mathcal{F})$, the relative Lie algebra cohomology $H^ullet(\mathfrak{g},\mathfrak{h})$, and the de Rham cohomology $H_{dR}^ullet(M)$. It identifies the second page $E_2^{p,q}$ as $H^p(M_2/\mathcal{F}_2, H^q(\mathfrak{g},\mathfrak{h}))^{\tilde{N}/N_0}$ (with equivalent formulations via the auxiliary cover $M_1$), and demonstrates how coverings and a blow-up construction make the framework applicable even when orbit spaces are singular or actions are not equidimensional. The work also derives significant simplifications in special cases (e.g., simply connected $M$ or trivial normalizer action), and provides Gysin and Wang-type exact sequences as concrete tools for computations. Finally, it uses a blow-up technique to exclude certain group actions on low-dimensional manifolds (notably $\mathbb{S}^3$-actions on $4$- and $5$-manifolds) and connects these obstructions to curvature questions, highlighting the practical impact for topology and geometric analysis.

Abstract

In this article we provide a version of the Leray-Serre spectral sequence for equidimensional (i.e. smooth with all orbits of the same dimension) actions of compact connected Lie groups on compact manifolds. The main part of this article consists of the proof of the description of the second page of said spectral sequence. This description provides a link between the cohomology of the orbit space (basic cohomology of the foliation by orbits) the Lie algebra cohomology of the appropriate pair $(\mathfrak{g},\mathfrak{h})$ representing the cohomology of a generic orbit and the de Rham cohomology of the manifold. Due to the somewhat technical nature of the general description we have provided in the penultimate section a thorough study of special cases in which the sequence can be greatly simplified. In particular, vast simplifications can be obtained if the manifold $M$ on which the group acts is assumed to be simply connected or if the acting Lie group has some nice properties. In the final section, we show how to use a blow up process to use our sequence when the action is not equidimensional. We apply this method to give a topological obstruction to the existence Lie group actions on certain manifolds.

A Spectral Sequence for Equidimensional Actions of Compact Lie Groups

TL;DR

The paper develops a Leray–Serre–type spectral sequence for equidimensional actions of a connected compact Lie group on a compact manifold , bridging the basic cohomology of the orbit foliation , the relative Lie algebra cohomology , and the de Rham cohomology . It identifies the second page as (with equivalent formulations via the auxiliary cover ), and demonstrates how coverings and a blow-up construction make the framework applicable even when orbit spaces are singular or actions are not equidimensional. The work also derives significant simplifications in special cases (e.g., simply connected or trivial normalizer action), and provides Gysin and Wang-type exact sequences as concrete tools for computations. Finally, it uses a blow-up technique to exclude certain group actions on low-dimensional manifolds (notably -actions on - and -manifolds) and connects these obstructions to curvature questions, highlighting the practical impact for topology and geometric analysis.

Abstract

In this article we provide a version of the Leray-Serre spectral sequence for equidimensional (i.e. smooth with all orbits of the same dimension) actions of compact connected Lie groups on compact manifolds. The main part of this article consists of the proof of the description of the second page of said spectral sequence. This description provides a link between the cohomology of the orbit space (basic cohomology of the foliation by orbits) the Lie algebra cohomology of the appropriate pair representing the cohomology of a generic orbit and the de Rham cohomology of the manifold. Due to the somewhat technical nature of the general description we have provided in the penultimate section a thorough study of special cases in which the sequence can be greatly simplified. In particular, vast simplifications can be obtained if the manifold on which the group acts is assumed to be simply connected or if the acting Lie group has some nice properties. In the final section, we show how to use a blow up process to use our sequence when the action is not equidimensional. We apply this method to give a topological obstruction to the existence Lie group actions on certain manifolds.

Paper Structure

This paper contains 20 sections, 41 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Foliations
  4. Lie algebras and Lie groups
  5. Classification of orbits for $SU(2)$-actions
  6. Proof of the Main Theorem
  7. A Choice of Riemannian Metric
  8. A Simplifying Lift for Equidimensional Actions
  9. Construction and the First Page of the Spectral Sequence
  10. The second page of the spectral sequence
  11. Special Cases, Examples and Applications
  12. Under what topological conditions can the coverings be ommited?
  13. For what groups can passing to the coverings be ommitted?
  14. Gysin long exact sequences
  15. Wang long exact sequences
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $M^{n}$ be a compact connected manifold with an equidimensional action of a compact connected Lie group $G$. Under the above notation, there is a spectral sequence $E^{p,q}_r$ converging to $H^{\bullet}_{dR}(M)$ with: Where $(M_1,\mathcal{F}_1)$ (resp. $(M_2,\mathcal{F}_2)$) are foliated manifolds covering $(M,\mathcal{F})$ (with $\mathcal{F}$ given by the orbits of the action) with correspon

Theorems & Definitions (80)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 70 more