Compact quotients of homogeneous spaces and homotopy theory of sphere bundles

A reductive homogeneous space $G\mathrm{/}H$ is always diffeomorphic to the normal bundle of an orbit of a maximal compact subgroup of $G$. We prove that if $G\mathrm{/}H$ 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 $\mathrm{O}(n+1,\mathbb{C})\mathrm{/}\mathrm{O}(n,\mathbb{C})$ for all $n\mathrm{\notin }\{1,3,7\}$, or $\mathrm{S}\mathrm{L}(n,\mathbb{R})\mathrm{/}\mathrm{S}\mathrm{L}(m,\mathbb{R})$ 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 $2^{\lfloor q\mathrm{/}2\rfloor }$.