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Compact quotients of homogeneous spaces and homotopy theory of sphere bundles

Fanny Kassel, Yosuke Morita, Nicolas Tholozan

TL;DR

This work establishes a homotopy-theoretic obstruction to the existence of compact quotients for reductive homogeneous spaces by showing that any such quotient forces the sphere bundle of the normal bundle to the maximal compact orbit to be fiber-homotopically trivial. The obstruction is implemented via topological KO/KU/KSp theory and the reduced J-group, with Adams operations linking vector-bundle data to divisibility constraints. The authors derive numerous nonexistence results, notably for complex spheres, SL(p+q,𝕂)/SL(p,𝕂) across 𝕂=ℝ,ℂ,ℍ, and a broad family of indefinite Grassmannians, thereby solving longstanding Kobayashi conjectures in many cases. They further connect the obstruction to a Geometric Fibration viewpoint, showing that compact quotients imply, and possibly are implied by, local geometric fibrations of G/H, with tangential homogeneous spaces playing a central role. Overall, the paper advances a cohesive K-theoretic framework to distinguish reductive homogeneous spaces that can and cannot admit compact quotients, with wide implications for rigidity, dynamics, and geometric structure theory.

Abstract

A reductive homogeneous space $G/H$ is always diffeomorphic to the normal bundle of an orbit of a maximal compact subgroup of $G$. We prove that if $G/H$ admits compact quotients, then the sphere bundle associated to this normal bundle is fiber-homotopically trivial. We deduce that many reductive homogeneous spaces do not admit compact quotients, such as the complex spheres $\mathrm{O}(n+1,\mathbb{C})/\mathrm{O}(n,\mathbb{C})$ for all $n \notin \{1,3,7\}$, or $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(m,\mathbb{R})$ for all $n>m>1$, which solves conjectures of T. Kobayashi from the early 1990s. We also prove that if the pseudo-Riemannian hyperbolic space $\mathbf{H}^{p,q}$ of signature $(p,q)$ admits compact quotients, then $p$ must be divisible by at least $2^{\lfloor q/2\rfloor}$.

Compact quotients of homogeneous spaces and homotopy theory of sphere bundles

TL;DR

This work establishes a homotopy-theoretic obstruction to the existence of compact quotients for reductive homogeneous spaces by showing that any such quotient forces the sphere bundle of the normal bundle to the maximal compact orbit to be fiber-homotopically trivial. The obstruction is implemented via topological KO/KU/KSp theory and the reduced J-group, with Adams operations linking vector-bundle data to divisibility constraints. The authors derive numerous nonexistence results, notably for complex spheres, SL(p+q,𝕂)/SL(p,𝕂) across 𝕂=ℝ,ℂ,ℍ, and a broad family of indefinite Grassmannians, thereby solving longstanding Kobayashi conjectures in many cases. They further connect the obstruction to a Geometric Fibration viewpoint, showing that compact quotients imply, and possibly are implied by, local geometric fibrations of G/H, with tangential homogeneous spaces playing a central role. Overall, the paper advances a cohesive K-theoretic framework to distinguish reductive homogeneous spaces that can and cannot admit compact quotients, with wide implications for rigidity, dynamics, and geometric structure theory.

Abstract

A reductive homogeneous space is always diffeomorphic to the normal bundle of an orbit of a maximal compact subgroup of . We prove that if admits compact quotients, then the sphere bundle associated to this normal bundle is fiber-homotopically trivial. We deduce that many reductive homogeneous spaces do not admit compact quotients, such as the complex spheres for all , or for all , which solves conjectures of T. Kobayashi from the early 1990s. We also prove that if the pseudo-Riemannian hyperbolic space of signature admits compact quotients, then must be divisible by at least .
Paper Structure (64 sections, 45 theorems, 286 equations)

This paper contains 64 sections, 45 theorems, 286 equations.

Table of Contents

  1. Introduction
  2. Problem and main theorem
  3. Topological K-Theory and applications
  4. Some important special cases
  5. Pseudo-Riemannian hyperbolic spaces
  6. Complex spheres
  7. The homogeneous space SL(p+q,R)/SL(p,R)
  8. Motivation: the Geometric fibration conjecture
  9. Structure of the paper
  10. Acknowledgements
  11. Reminders: reductive homogeneous spaces
  12. Reductive Lie groups and reductive homogeneous spaces
  13. Reductive symmetric spaces and associated pairs
  14. A sufficient condition for fiber-homotopy equivalence
  15. Fiber-homotopical triviality of the normal sphere bundle
  16. ...and 49 more sections

Key Result

Theorem 1.3

If the reductive homogeneous space $G/H$ admits compact quotients, then $S(N)$ is fiber-homotopically trivial.

Theorems & Definitions (137)

  • Conjecture 1.2: Kobayashi
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • Corollary 1.11
  • ...and 127 more