Jordan–Schönflies theorem

Jordan–Schönflies theorem

In mathematics, the Jordan–Schönflies theorem, or simply the Schönflies theorem, of geometric topology is a sharpening of the Jordan curve theorem.

Formulation

It states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle. Putting this more precisely: The plane can be mapped onto itself through a continuous bijection whose inverse is also continuous (the definition of a homeomorphism) to make the simple closed curve become a circle; the inside and outside of the curve become those of the circle.

Such a theorem is only valid in two dimensions. In three dimensions there are counterexamples such as Alexander's horned sphere. Although they separate space into two regions, those regions are so twisted and knotted that they are not homeomorphic to the inside and outside of a normal sphere.

Generalizations

There does exist a higher-dimensional generalization due to Morton Brown and independently Barry Mazur with Marston Morse, which is also called the Schönflies theorem. It states that, if an ("n" − 1)-dimensional sphere "S" is embedded into the "n"-dimensional sphere "Sn" in a locally flat way (that is, the embedding extends to that of a thickened sphere), then the pair ("Sn", "S") is homeomorphic to the pair ("Sn", "S""n"−1), where "S""n"−1 is the equator of the "n"-sphere. Brown and Mazur received the Veblen Prize for their contributions.

The Schonflies problem can be posed in categories other than the topologically locally flat category, i.e. does a smoothly (piecewise-linearly) embedded "n-1"-sphere in the "n"-sphere bound a smooth (piecewise-linear) "n"-ball? For "n" = 4, the problem is still open for both categories. See Mazur manifold.

References

*Brown, Morton (1960), A proof of the generalized Schoenflies theorem. "Bull. Amer. Math. Soc.", vol. 66, pp. 74–76. MathSciNet| id = 0117695
*Mazur, Barry, On embeddings of spheres., "Bull. Amer. Math. Soc." 65 1959 59--65. MathSciNet| id = MR0117693
*Morse, Marston, A reduction of the Schoenflies extension problem., "Bull. Amer. Math. Soc." 66 1960 113--115. MathSciNet| id = 0117694


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