Atoms of Compacta on Closed Surfaces

TL;DR

This paper introduces a new equivalence relation on compact sets on closed surfaces, defining atoms as core components with a Peano quotient, and explores their behavior under branched coverings, with examples and counterexamples.

Contribution

It establishes the core decomposition of compact sets on closed surfaces into atoms with Peano quotients and analyzes their properties under branched coverings.

Findings

Atoms are continua and form a core decomposition refining other decompositions.

Under branched coverings, atoms map into atoms, with surjective mappings for coverings.

The theory does not extend to higher-dimensional manifolds, as shown by a counterexample.

Abstract

For any compact set $K$ lying on a closed surface $\mathcal{S}$ we introduce a closed equivalence relation $\sim$, called the {\em Sch\"onflies 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~$\mathbb{R}^3$.