Diffeology and Arithmetic of Irrational Tori
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908742704955392 |
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| author | Iglesias-Zemmour, Patrick |
| author_facet | Iglesias-Zemmour, Patrick |
| contents | The irrational torus, $\mathrm{T}_α$, originally introduced as a geometric model for quasicrystals, is a foundational object in the theory of diffeology. This paper, after recalling its main algebraic properties, provides a comprehensive analysis of a new geometric invariant for this singular space: the group of flows, $\mathbf{Fl}(\mathrm{T}_α, \mathbf{R})$. This invariant, which is trivial for all manifolds, arises as the core of the obstruction to the de Rham theorem in the diffeological setting. We provide a complete computation and geometric interpretation of this group, proving the isomorphism $\mathbf{Fl}(\mathrm{T}_α, \mathbf{R}) \simeq \mathbf{R} \times \mathrm{coker}(Δ_α)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_07460 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Diffeology and Arithmetic of Irrational Tori Iglesias-Zemmour, Patrick Mathematical Physics 58A40, 55R15, 37E10, 11J71, 57R30 The irrational torus, $\mathrm{T}_α$, originally introduced as a geometric model for quasicrystals, is a foundational object in the theory of diffeology. This paper, after recalling its main algebraic properties, provides a comprehensive analysis of a new geometric invariant for this singular space: the group of flows, $\mathbf{Fl}(\mathrm{T}_α, \mathbf{R})$. This invariant, which is trivial for all manifolds, arises as the core of the obstruction to the de Rham theorem in the diffeological setting. We provide a complete computation and geometric interpretation of this group, proving the isomorphism $\mathbf{Fl}(\mathrm{T}_α, \mathbf{R}) \simeq \mathbf{R} \times \mathrm{coker}(Δ_α)$. |
| title | Diffeology and Arithmetic of Irrational Tori |
| topic | Mathematical Physics 58A40, 55R15, 37E10, 11J71, 57R30 |
| url | https://arxiv.org/abs/2508.07460 |