A topological necessary condition for the existence of compact 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-Ono with an elementary fact that certain two different Clifford-Klein forms have the same cohomology ring. We give some examples, SL(p+q, R)/SO(p, q) (p, q: odd) for instance, of homogeneous spaces which do not admit compact Clifford-Klein forms.

16 pages, revised following referee's comments, to appear in Journal of Differential Geometry

Cited by in corpus (7)

• Volume and non-existence of compact Clifford-Klein forms
• Homogeneous spaces of nonreductive type locally modelling no compact manifold
• A cohomological obstruction to the existence of compact Clifford-Klein forms
• Semisimple subalgebras in simple Lie algebras and a computational approach to the compact Clifford-Klein forms problem
• Proof of Kobayashi's rank conjecture on Clifford-Klein forms
• Cartan projections of some non-reductive subgroups and proper actions on homogeneous spaces
• On locally homogeneous pseudo-Riemannian compact einstein manifolds