A Discrete Proof of The General Jordan-Schoenflies Theorem

TL;DR

This paper presents a constructive, discrete proof of the general Jordan-Schoenflies theorem, demonstrating that locally flat embeddings of spheres in spheres decompose the space into two homeomorphic to n-balls, with implications for computational topology.

Contribution

It provides a novel discrete method-based constructive proof of the theorem, emphasizing computability and potential applications in design algorithms.

Findings

Proof confirms decomposition of n-sphere by locally flat embedded (n-1)-sphere

Method evaluates computational costs of homeomorphism operations

Extensions applicable to design algorithms with constructible homeomorphisms

Abstract

In the early 1960s, Brown and Mazur proved the general Jordan-Schoenflies theorem. This fundamental theorem states: If we embed an $(n-1)$ sphere $S^{(n-1)}$ locally flatly in an $n$ sphere $S^{n}$, then it decomposes $S^{n}$ into two components. In addition, the embedded $S^{(n-1)}$ is the common boundary of the two components and each component is homeomorphic to the $n$-ball.\newline This paper gives a constructive proof of the theorem using the discrete method. More specifically, we prove the equivalent statements: Let $M$ be an $n$-manifold, which is homeomorphic to $S^{n}$. Then, every $(n-1)$-manifold $S$, a submanifold with local flatness in $M$, decomposes the space $M$ into two components where each component is homeomorphic to an $n$-ball. The method was chosen in order to evaluate the computability and computational costs among operations between cells regarding homeomorphism. In addition, methods within the proof can be extended to applications in design algorithms under the assumption that homeomorphic mappings are constructible and computable. In this new revision, We add some new detailed discussions.