Classification of differentiable structures on the non-Hausdorff line with two origins
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929385672540160 |
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| author | Lysynskyi, Mykola Maksymenko, Sergiy |
| author_facet | Lysynskyi, Mykola Maksymenko, Sergiy |
| contents | We classify differentiable structures on a line $\mathbb{L}$ with two origins being a non-Hausdorff but $T_1$ one-dimensional manifold obtained by ``doubling'' $0$. For $k\in\mathbb{N}\cup\{\infty\}$ let $H$ be the group of homeomorphisms $h$ of $\mathbb{R}$ such that $h(0)=0$ and the restriction of $h$ to $\mathbb{R}\setminus0$ is a $\mathcal{C}^{k}$-diffeomorphism. Let also $D$ be the subgroup of $H$ consisting of $\mathcal{C}^{k}$-diffeomorphisms of $\mathbb{R}$ also fixing $0$. It is shown that there is a natural bijection between $\mathcal{C}^{k}$-structures on $\mathbb{L}$ (up to a $\mathcal{C}^{k}$-diffeomorphism fixing both origins) and double $D$-coset classes $D \setminus H / D = \{ D h D \mid h \in H\}$.
Moreover, the set of all $\mathcal{C}^{k}$-structures on $\mathbb{L}$ (up to a $\mathcal{C}^{k}$-diffeomorphism which may also exchange origins) are in one-to-one correspondence with the set of double $(D,\pm)$-coset classes $D \setminus H^{\pm} / D = \{ D h D \cup D h^{-1} D \mid h \in H\}$.
In particular, in contrast with the real line, the line with two origins $\mathbb{L}$ admits uncountably many pair-wise non-diffeomorphic $\mathcal{C}^{k}$-structures for each $k=1,2,\ldots,\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09576 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Classification of differentiable structures on the non-Hausdorff line with two origins Lysynskyi, Mykola Maksymenko, Sergiy Geometric Topology Algebraic Topology Differential Geometry Dynamical Systems General Topology 58A05, 57R30 We classify differentiable structures on a line $\mathbb{L}$ with two origins being a non-Hausdorff but $T_1$ one-dimensional manifold obtained by ``doubling'' $0$. For $k\in\mathbb{N}\cup\{\infty\}$ let $H$ be the group of homeomorphisms $h$ of $\mathbb{R}$ such that $h(0)=0$ and the restriction of $h$ to $\mathbb{R}\setminus0$ is a $\mathcal{C}^{k}$-diffeomorphism. Let also $D$ be the subgroup of $H$ consisting of $\mathcal{C}^{k}$-diffeomorphisms of $\mathbb{R}$ also fixing $0$. It is shown that there is a natural bijection between $\mathcal{C}^{k}$-structures on $\mathbb{L}$ (up to a $\mathcal{C}^{k}$-diffeomorphism fixing both origins) and double $D$-coset classes $D \setminus H / D = \{ D h D \mid h \in H\}$. Moreover, the set of all $\mathcal{C}^{k}$-structures on $\mathbb{L}$ (up to a $\mathcal{C}^{k}$-diffeomorphism which may also exchange origins) are in one-to-one correspondence with the set of double $(D,\pm)$-coset classes $D \setminus H^{\pm} / D = \{ D h D \cup D h^{-1} D \mid h \in H\}$. In particular, in contrast with the real line, the line with two origins $\mathbb{L}$ admits uncountably many pair-wise non-diffeomorphic $\mathcal{C}^{k}$-structures for each $k=1,2,\ldots,\infty$. |
| title | Classification of differentiable structures on the non-Hausdorff line with two origins |
| topic | Geometric Topology Algebraic Topology Differential Geometry Dynamical Systems General Topology 58A05, 57R30 |
| url | https://arxiv.org/abs/2406.09576 |