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A Discrete Proof of The General Jordan-Schoenflies Theorem | | Li Chen
; Steven G. Krantz
; | | Date: |
21 Apr 2015 | | Abstract: | In this paper we give a discrete proof of the general Jordan-Schoenflies
Theorem. The classical Jordan-Schoenflies Theorem states that a simple closed
curve in the two-dimensional sphere $S^2$ separates the space into two
connected components where each component is homeomorphic to an open disk. The
common boundary of these two components is this closed curve.
The general Jordan-Schoenflies Theorem extends this property to any
dimension: Every $(n-1)$-submanifold $S$ that is homeomorphic to a sphere, and
is a submanifold with a suitable local flatness condition, in an $n$-manifold
$M$ which is homeomorphic to an $n$-sphere, decomposes the space $M$ into two
components, and each of the components is homeomorphic to an $n$-cell. In other
words, embedding an $(n-1)$-sphere $S^{(n-1)}$ nicely in an $n$-sphere $S^{n}$,
decomposes the space into two components and the embedded $S^{(n-1)}$ is their
common boundary. Each of the two components is homeomorphic to the $n$-ball. In
the early 1960s, Brown cite{Bro} and Mazur cite{Maz} proved this fundamental
theorem.
In the present paper, we provide a constructive proof of the theorem using
the discrete method. Our proof can also be used to design algorithms for
applications. | | Source: | arXiv, 1504.5263 | | Services: | Forum | Review | PDF | Favorites | |
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