Recognizing indecomposable subcontinua of surfaces from their complements

TL;DR

This paper provides criteria to identify indecomposable subcontinua in closed surfaces by analyzing the behavior of their complementary domains, introducing the double-pass condition as a key indicator.

Contribution

It establishes new theorems for recognizing indecomposable continua in surfaces and proposes the double-pass condition as an equivalent criterion for indecomposability.

Findings

A continuum is either indecomposable or the union of two indecomposables under certain boundary conditions.

The double-pass condition is conjectured to characterize indecomposability.

Proved theorems apply to continua not forming the boundary of their complementary domain.

Abstract

We prove two theorems which allow one to recognize indecomposable subcontinua of closed surfaces without boundary. If $X$ is a subcontinuum of a closed surface $S$, we call the components of $S \setminus X$ the complementary domains of $X$. We prove that a continuum is either indecomposable or the union of two indecomposable continua whenever it has a sequence of distinct complementary domains whose boundaries limit to the continuum in the Hausdorff metric. We define a slightly stronger condition on the complementary domains of a continuum, called the double-pass condition, which we conjecture is equivalent to indecomposability of the continuum. We prove that this is so for continua which are not the boundary of one of their complementary domains.