Differentiable structures on a union of two open sets
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908438020227072 |
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| author | Lysynskyi, Mykola Maksymenko, Sergiy |
| author_facet | Lysynskyi, Mykola Maksymenko, Sergiy |
| contents | In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold $\mathbb{L}$ called the line with two origins which is obtained by gluing two copies of the real line $\mathbb{R}$ via the identity homeomorphism of $\mathbb{R}\setminus 0$.
Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold $\mathbb{Y}$ (called letter "$Y$") obtained by gluing two copies of $\mathbb{R}$ via the identity map of positive reals. It turns out that, in contrast to the real line, for every $r=1,\ldots,\infty$, both manifolds $\mathbb{L}$ and $\mathbb{Y}$ admit uncountably many pair-wise non-diffeomorphic $\mathcal{C}^{k}$-structures.
We also observe that the proofs of these classifications are very similar. This allows to formalize the arguments and extend them to a certain general statement about arrows in arbitrary categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05156 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Differentiable structures on a union of two open sets Lysynskyi, Mykola Maksymenko, Sergiy Differential Geometry Algebraic Geometry Algebraic Topology Category Theory 58A05, 57R30 In a recent paper the authors classified differentiable structures on the non-Hausdorff one-dimensional manifold $\mathbb{L}$ called the line with two origins which is obtained by gluing two copies of the real line $\mathbb{R}$ via the identity homeomorphism of $\mathbb{R}\setminus 0$. Here we give a classification of differentiable structures on another non-Hausdorff one-dimensional manifold $\mathbb{Y}$ (called letter "$Y$") obtained by gluing two copies of $\mathbb{R}$ via the identity map of positive reals. It turns out that, in contrast to the real line, for every $r=1,\ldots,\infty$, both manifolds $\mathbb{L}$ and $\mathbb{Y}$ admit uncountably many pair-wise non-diffeomorphic $\mathcal{C}^{k}$-structures. We also observe that the proofs of these classifications are very similar. This allows to formalize the arguments and extend them to a certain general statement about arrows in arbitrary categories. |
| title | Differentiable structures on a union of two open sets |
| topic | Differential Geometry Algebraic Geometry Algebraic Topology Category Theory 58A05, 57R30 |
| url | https://arxiv.org/abs/2507.05156 |