Compact quotients of homogeneous spaces and homotopy theory of sphere bundles
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866908755399016448 |
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| author | Kassel, Fanny Morita, Yosuke Tholozan, Nicolas |
| author_facet | Kassel, Fanny Morita, Yosuke Tholozan, Nicolas |
| contents | 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 $2^{\lfloor q/2\rfloor}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05857 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Compact quotients of homogeneous spaces and homotopy theory of sphere bundles Kassel, Fanny Morita, Yosuke Tholozan, Nicolas Geometric Topology Algebraic Topology Differential Geometry 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 $2^{\lfloor q/2\rfloor}$. |
| title | Compact quotients of homogeneous spaces and homotopy theory of sphere bundles |
| topic | Geometric Topology Algebraic Topology Differential Geometry |
| url | https://arxiv.org/abs/2601.05857 |