Compact quotients of homogeneous spaces and homotopy theory of sphere bundles

TL;DR

This paper investigates conditions under which reductive homogeneous spaces admit compact quotients, revealing topological obstructions via homotopy theory and solving longstanding conjectures.

Contribution

It establishes that sphere bundles over such spaces are fiber-homotopically trivial when compact quotients exist, and proves new divisibility conditions for hyperbolic space quotients.

Findings

Many homogeneous spaces do not admit compact quotients.

Sphere bundles are fiber-homotopically trivial in these cases.

Divisibility conditions on parameters for hyperbolic space quotients.

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 $2^{\lfloor q/2\rfloor}$.