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Yosuke Morita

Publications and source records attributed to Yosuke Morita.

9 recordsLinked to original sources
arXiv2026

Compact quotients of homogeneous spaces and homotopy theory of sphere bundles

A reductive homogeneous space $G/H$ is always diffeomorphic to the normal bundle of an orbit of a maximal compact subgroup of $G$. We prove that if $G/H$ admits compact quotients, then the sphere bundle associated to this normal bundle is fiber-homotopically trivial. We deduce that many reductive homogeneous spaces do not admit compact quotients, such as the complex spheres $\mathrm{O}(n+1,\mathbb{C})/\mathrm{O}(n,\mathbb{C})$ for all $n \notin \{1,3,7\}$, or $\mathrm{SL}(n,\mathbb{R})/\mathrm{SL}(m,\mathbb{R})$ for all $n>m>1$, which solves conjectures of T. Kobayashi from the early 1990s. We also prove that if the pseudo-Riemannian hyperbolic space $\mathbf{H}^{p,q}$ of signature $(p,q)$ admits compact quotients, then $p$ must be divisible by at least $2^{\lfloor q/2\rfloor}$.

math.GT↗
arXiv2025

Exotic proper actions on homogeneous spaces via convex cocompact representations

We construct a series of homogeneous spaces G/H of reductive type which admit proper actions of discrete subgroups of G isomorphic to cocompact lattices of O(n,1) (n=2,3,4) but do not admit proper actions of non-compact semisimple subgroups of G. The existence of such homogeneous spaces was previously not known even for n=2. Our construction of proper actions of discrete subgroups is based on Gu\'eritaud-Kassel's work on convex cocompact subgroups of O(n,1) and Danciger-Gu\'eritaud-Kassel's work on right-angled Coxeter groups. On the other hand, the non-existence of proper actions of non-compact semisimple subgroups is proved by the theory of nilpotent orbits and elementary combinatorics.

math.GR↗
arXiv2023

Conley index theory without index pairs. I: The point-set level theory

We propose a new framework for Conley index theory. The main feature of our approach is that we do not use the notion of index pairs. We introduce, instead, the notions of compactifiable subsets and index neighbourhoods, and formulate and prove basic results in Conley index theory using these notions. We treat both the discrete time case and the continuous time case.

math.DS↗
arXiv2022

Cartan projections of some non-reductive subgroups and proper actions on homogeneous spaces

Kobayashi [Duke Math. J. (1992)] gave a necessary condition for the existence of compact Clifford-Klein forms in terms of Cartan projections and non-compact dimensions of reductive subgroups. We extend his method to non-reductive subgroups, and give some examples of homogeneous spaces of reductive type that do not admit compact Clifford-Klein forms by comparing Cartan projections and non-compact dimensions of reductive subgroups with those of non-reductive subgroups.

math.DG↗
arXiv2019

Proof of Kobayashi's rank conjecture on Clifford-Klein forms

T. Kobayashi conjectured in the 36th Geometry Symposium in Japan (1989) that a homogeneous space G/H of reductive type does not admit a compact Clifford-Klein form if rank G - rank K < rank H - rank K_H. We solve this conjecture affirmatively. We apply a cohomological obstruction to the existence of compact Clifford-Klein forms proved previously by the author, and use the Sullivan model for a reductive pair due to Cartan-Chevalley-Koszul-Weil.

math.GT↗
arXiv2016

A cohomological obstruction to the existence of compact Clifford-Klein forms

In this paper, we continue the study of the existence problem of compact Clifford-Klein forms from a cohomological point of view, which was initiated by Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an obstruction to the existence of compact Clifford-Klein forms by relating a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology with an upper-bound estimate for cohomological dimensions of discontinuous groups. From this obstruction, we derive some examples, e.g. $\mathrm{SO}_0(p+r, q)/(\mathrm{SO}_0(p,q) \times \mathrm{SO}(r))$ $(p,q,r \geq 1, \ q:\text{odd})$ and $\mathrm{SL}(p+q, \mathbb{C})/\mathrm{SU}(p,q)$ $(p,q \geq 1)$, of a homogeneous space that does not admit a compact Clifford-Klein form. To construct these examples, we apply H. Cartan's theorem on relative Lie algebra cohomology of reductive pairs and the theory of $ε$-families of semisimple symmetric pairs.

math.GT↗
arXiv2015

Homogeneous spaces of nonreductive type locally modelling no compact manifold

We give necessary conditions for the existence of a compact manifold locally modelled on a given homogeneous space, which generalize some earlier results, in terms of relative Lie algebra cohomology. Applications include both reductive and nonreductive cases. For example, we prove that there does not exist a compact manifold locally modelled on a positive dimensional coadjoint orbit of a real linear solvable algebraic group.

math.DG↗
arXiv2014

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

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.

math.DG↗