Séminaire Géométrie Topologie Dynamique
The existence problem of compact quotients of homogeneous spaces
Nov. 2023
Intervenant : Yosuke Morita
Institution : Kyushu University
Heure : 14h00 - 15h00
Lieu : 2L8

Let $G/H$ be a homogeneous space. If a discrete subgroup of G acts properly and freely on $G/H$, the quotient space becomes a manifold locally modelled on G/H and is called a Clifford-Klein form. For example, when $G/H$ is (the universal cover of) the pseudo-Riemannian hyperbolic space $\mathbb{H}^{p, q} = PO(p, q+1) / P(O(p, q) \times O(1))$, Clifford-Klein forms are precisely complete pseudo-Riemannian manifolds of signature $(p, q)$ with constant negative sectional curvature. In this talk, I plan to explain my recent work (joint with Fanny Kassel and Nicolas Tholozan) on a new necessary condition for the existence of compact Clifford-Klein forms, which is formulated in terms of algebraic topology of sphere bundles. Our result and J. Frank Adams's work in 1960s together imply that $\mathbb{H}^{p, q}$ does not admit compact Clifford-Klein forms in most cases. For instance, a complete pseudo-Riemannian manifold of signature (p, 20) with constant negative sectional curvature is never compact except possibly when $p$ is a multiple of 2048.

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