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Positive curvature and rational ellipticity in cohomogeneity three

Elahe Khalili Samani, Marco Radeschi

TL;DR

The paper proves that a closed, simply connected, positively curved manifold $M$ with a cohomogeneity-three action and a boundary-free quotient $M/G$ is rationally elliptic, extending Bott–Grove–Halperin-type results to new symmetry settings. It develops two complementary pathways: a rational-homotopy approach that leverages Sullivan minimal models and cohomological constraints, and a double-disk-bundle decomposition strategy that reduces the problem to rationally elliptic pieces. A thorough analysis of the quotient’s singular strata, aided by branched covers and quotient-geodesic techniques, yields a finite list of possible configurations, all of which are shown to lead to rational ellipticity. The results thus generalize prior findings for homogeneous, cohomogeneity-one, and cohomogeneity-two actions to the cohomogeneity-three setting, with potential implications for understanding topology under symmetry in positively curved manifolds.

Abstract

We prove that a closed, simply connected, positively curved, cohomogeneity-three manifold whose quotient space has no boundary is rationally elliptic, thus providing a generalization of similar results regarding rational ellipticity of homogeneous, cohomogeneity-one, and almost non-negatively curved cohomogeneity-two manifolds.

Positive curvature and rational ellipticity in cohomogeneity three

TL;DR

The paper proves that a closed, simply connected, positively curved manifold with a cohomogeneity-three action and a boundary-free quotient is rationally elliptic, extending Bott–Grove–Halperin-type results to new symmetry settings. It develops two complementary pathways: a rational-homotopy approach that leverages Sullivan minimal models and cohomological constraints, and a double-disk-bundle decomposition strategy that reduces the problem to rationally elliptic pieces. A thorough analysis of the quotient’s singular strata, aided by branched covers and quotient-geodesic techniques, yields a finite list of possible configurations, all of which are shown to lead to rational ellipticity. The results thus generalize prior findings for homogeneous, cohomogeneity-one, and cohomogeneity-two actions to the cohomogeneity-three setting, with potential implications for understanding topology under symmetry in positively curved manifolds.

Abstract

We prove that a closed, simply connected, positively curved, cohomogeneity-three manifold whose quotient space has no boundary is rationally elliptic, thus providing a generalization of similar results regarding rational ellipticity of homogeneous, cohomogeneity-one, and almost non-negatively curved cohomogeneity-two manifolds.

Paper Structure

This paper contains 17 sections, 24 theorems, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation on isometric group actions
  4. Alexandrov spaces with positive curvature
  5. Classification of singular points in the quotient of a cohomogeneity-three group action
  6. Quotient geodesics
  7. Branched covers of Alexandrov spaces along singular cycles
  8. Techniques for identifying rational ellipticity
  9. Rational homotopy theory
  10. Restricting possible dimensions and groups in special cases
  11. Quotients with two non-orbifold points
  12. Quotients with one non-orbifold point
  13. Double disk bundle decomposition
  14. Applying the results of Part \ref{['P:techniques']} to the quotient space $M/G$
  15. The structure of the singular strata of the quotient space
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose $M$ is a closed, simply connected, positively curved Riemannian manifold which admits a cohomogeneity-three action by a closed Lie group $G$ with $\partial (M/G)=\emptyset$. Then $M$ is rationally elliptic.

Figures (1)

  • Figure 1: Double disk bundle decompositions. Dashed lines (resp. the hollow point in the first picture) represent curves (resp. point) chosen in the regular part.

Theorems & Definitions (55)

  • Theorem 1
  • Remark 1.1
  • Example 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • ...and 45 more