Atoms of Compacta on Closed Surfaces

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 $\epsilon >0$ only finitely many of them have diameter greater than $\epsilon$. 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 $f:{\mathcal{S}}^{\ast }\to \mathcal{S}$ from a closed surface ${\mathcal{S}}^{\ast }$ 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~${\mathbb{R}}^3$.

26 pages, 13 figures