Atoms of Compacta on Closed Surfaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914462559109120 |
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| author | Luo, Jun Thuswaldner, Joerg Yao, Xiao-Ting Zhang, Shuqin |
| author_facet | Luo, Jun Thuswaldner, Joerg Yao, Xiao-Ting Zhang, Shuqin |
| contents | For any compact set $K$ lying on a closed surface $\mathcal{S}$ we introduce a closed equivalence relation $\sim$, called the {\em Schönflies equivalence} on $K$. We show that every class $[x]_\sim$ of $\sim$ is a continuum and that the resulting quotient space $K\!/\!\sim$ is a {\em Peano compactum}. By definition, all components of a Peano compactum are locally connected and for any $\varepsilon>0$ only finitely many of them have diameter greater than $\varepsilon$. The decomposition $\mathcal{D}_K=\{[x]_\sim: x\in K\}$ refines every other upper semicontinuous decomposition of $K$ into subcontinua that has a Peano compactum as its quotient space. In other words, $\mathcal{D}_K$ is the {\em core decomposition of $K$} with Peano quotient. The elements of $\mathcal{D}_K$ are called {\em atoms} of $K$. We also show that for any branched covering $f: \mathcal{S}^*\rightarrow \mathcal{S}$ from a closed surface $\mathcal{S}^*$ to $\mathcal{S}$, every atom of $f^{-1}(K)$ is sent into an atom of $K$. If $f$ is even a covering, it sends every atom of $f^{-1}(K)$ onto an atom of $K$. We illustrate our theory with examples and show that it cannot be generalized to $n$-manifolds with $n\ge 3$ by providing a detailed counterexample in~$\mathbb{R}^3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_27054 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Atoms of Compacta on Closed Surfaces Luo, Jun Thuswaldner, Joerg Yao, Xiao-Ting Zhang, Shuqin General Topology 54D05, 54H20, 37F45, 37E99 For any compact set $K$ lying on a closed surface $\mathcal{S}$ we introduce a closed equivalence relation $\sim$, called the {\em Schönflies equivalence} on $K$. We show that every class $[x]_\sim$ of $\sim$ is a continuum and that the resulting quotient space $K\!/\!\sim$ is a {\em Peano compactum}. By definition, all components of a Peano compactum are locally connected and for any $\varepsilon>0$ only finitely many of them have diameter greater than $\varepsilon$. The decomposition $\mathcal{D}_K=\{[x]_\sim: x\in K\}$ refines every other upper semicontinuous decomposition of $K$ into subcontinua that has a Peano compactum as its quotient space. In other words, $\mathcal{D}_K$ is the {\em core decomposition of $K$} with Peano quotient. The elements of $\mathcal{D}_K$ are called {\em atoms} of $K$. We also show that for any branched covering $f: \mathcal{S}^*\rightarrow \mathcal{S}$ from a closed surface $\mathcal{S}^*$ to $\mathcal{S}$, every atom of $f^{-1}(K)$ is sent into an atom of $K$. If $f$ is even a covering, it sends every atom of $f^{-1}(K)$ onto an atom of $K$. We illustrate our theory with examples and show that it cannot be generalized to $n$-manifolds with $n\ge 3$ by providing a detailed counterexample in~$\mathbb{R}^3$. |
| title | Atoms of Compacta on Closed Surfaces |
| topic | General Topology 54D05, 54H20, 37F45, 37E99 |
| url | https://arxiv.org/abs/2603.27054 |