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Atoms of Compacta on Closed Surfaces

Jun Luo, Joerg Thuswaldner, Xiao-Ting Yao, Shuqin Zhang

Abstract

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~$\mathbb{R}^3$.

Atoms of Compacta on Closed Surfaces

Abstract

For any compact set lying on a closed surface we introduce a closed equivalence relation , called the {\em Schönflies equivalence} on . We show that every class of is a continuum and that the resulting quotient space is a {\em Peano compactum}. By definition, all components of a Peano compactum are locally connected and for any only finitely many of them have diameter greater than . The decomposition refines every other upper semicontinuous decomposition of into subcontinua that has a Peano compactum as its quotient space. In other words, is the {\em core decomposition of } with Peano quotient. The elements of are called {\em atoms} of . We also show that for any branched covering from a closed surface to , every atom of is sent into an atom of . If is even a covering, it sends every atom of onto an atom of . We illustrate our theory with examples and show that it cannot be generalized to -manifolds with by providing a detailed counterexample in~.

Paper Structure

This paper contains 7 sections, 26 theorems, 15 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Statement of the main results
  3. Preliminaries
  4. Existence of the core decomposition and its characterization by Schönflies equivalence
  5. How atoms are mapped under branched coverings
  6. Examples for determining the core decomposition
  7. Atomic decompositions of compacta in $\mathbb{R}^3$

Key Result

Theorem 2.3

Let $K\subset \mathcal{S}$. Then the core decomposition of $K$ exists and is given by $\mathcal{D}_K$.

Figures (13)

  • Figure 1: A Quadrialteral with marked edges $I_1$ and $I_2$.
  • Figure 2: A patch of the brick-wall tiling $\mathcal{T}_\varepsilon$.
  • Figure 3: Relative locations of $x_0,x_1$ and $P_0,P_1,P_2$.
  • Figure 4: An image of $Q$ and $B$, the latter of which is shaded.
  • Figure 5: Relative locations of $\alpha_0, \beta_0$ and $z_1\in \gamma_1$ in $Q$.
  • ...and 8 more figures

Theorems & Definitions (70)

  • Definition 1.1: $F$-compactum and Peano compactum; cf. LLY-2019
  • Definition 1.2: Some kinds of decompositions
  • Remark 1.3
  • Definition 2.1: The Schönflies relation on surfaces
  • Definition 2.2: Schönflies equivalence on surfaces
  • Theorem 2.3
  • Corollary 2.4
  • Theorem 2.5
  • Lemma 3.1: Cut Wire Theorem, see e.g. Nadler92
  • Lemma 3.2: Torhorst Theorem, see e.g. Kuratowski68
  • ...and 60 more