A topological necessary condition for the existence of compace Clifford-Klein forms

Abstract

We provide a necessary condition for the existence of a compact Clifford–Klein form of a given homogeneous space of reductive type. The key to the proof is to combine a result of Kobayashi and Ono with the observation that a fiber bundle with contractible fiber induces an isomorphism between the cohomology rings of the total space and the base space. We give some examples: $SL(p + q,\mathbb{R})/SO(p, q)(p, q : \mathrm{odd})$, for instance—of homogeneous spaces that do not admit compact Clifford–Klein forms.