A Discrete Proof of The General Jordan-Schoenflies Theorem

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.