A cohomological obstruction to the existence of Clifford-Klein forms

A Clifford-Klein form is a quotient of a homogeneous space G/H by a discrete subgroup of G acting properly and freely on G/H. It naturally admits a structure of a manifold locally modeled on G/H. Comparing relative Lie algebra cohomology and de Rham cohomology, we give a new obstruction to the existence of compact Clifford-Klein forms of a given homogeneous space. As a corollary, we see that every complete pseudo-Riemannian manifold of signature (p, q) with positive constant sectional curvature is noncompact if p, q 0, q: odd.