Obstructions to the existence of compact Clifford-Klein forms for tangential symmetric spaces

TL;DR

This paper establishes new obstructions to the existence of compact Clifford-Klein forms for tangential symmetric spaces, identifying specific cases where such forms cannot exist and narrowing the possibilities for classical semisimple symmetric spaces.

Contribution

It provides easy-to-check necessary conditions for the existence of compact quotients in the tangential setting, advancing understanding of obstructions in symmetric spaces.

Findings

Identified new obstructions to compact Clifford-Klein forms in tangential symmetric spaces.

Showed only two types of irreducible classical semisimple symmetric spaces remain unclassified.

Connected obstructions to various mathematical fields like vector bundle trivializability and polynomial zero points.

Abstract

For a homogeneous space $G/H$ of reductive type, we consider the tangential homogeneous space $G_\theta/H_\theta$. In this paper, we give obstructions to the existence of compact Clifford-Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford-Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford-Klein forms. The existence problem of compact Clifford-Klein forms for homogeneous spaces of reductive type, which was initiated by T. Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by T. Kobayashi and T. Yoshino. In concrete examples, further works are needed to verify Kobayashi-Yoshino's condition by direct calculations. In this paper, some easy-to-check necessary conditions(=obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi-Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz-Radon number and Pfister's theorem (the existence problem of common zero points of polynomials of odd degree).