Schoenflies problem

The Schoenflies problem asks whether an embedded sphere in a higher-dimensional sphere bounds two regions homeomorphic to balls. Its answer depends on the dimension and regularity of the embedding.

Topological embedding: A continuous map that is a homeomorphism between its domain and its image. The problem concerns spheres placed inside larger spheres without assuming they are smoothly embedded.
Schoenflies theorem: The theorem that a simple closed curve in the plane bounds a region homeomorphic to a disk, with the outside also disk-like on the sphere. It gives the classical low-dimensional case of the problem.
Alexander horned sphere: A wild embedding of a two-sphere in three-dimensional space whose complement has a non-simply connected component. Its inside is not a ball, giving a canonical negative example for unrestricted embeddings.
Smooth Schoenflies conjecture: The conjecture that every smoothly embedded three-sphere in the four-sphere bounds two standard smooth four-balls. This dimension-four smooth version remains a central unresolved form of the problem.
n-sphere: The space of points at unit distance from the origin in Euclidean space of one higher dimension. These are the embedded hypersurfaces whose complementary regions the problem tests.
Wild embedding: An embedding that cannot be carried to a standard embedding by an ambient homeomorphism. Wildness can obstruct the ball conclusion even when the embedded object is a sphere.
Standard sphere embedding: The inclusion of a round sphere as the boundary of a round ball in a larger sphere. This model embedding has two complementary regions that are balls.
Exotic 4-ball: A smooth four-manifold homeomorphic but not known to be diffeomorphic to the standard four-ball. A nonstandard smooth ball would complicate the smooth conclusion sought in dimension four.
n-ball: A space homeomorphic to the closed unit ball in n-dimensional Euclidean space. The question is whether both sides of an embedded sphere have this topology.
Locally flat embedding: An embedding whose pair near each point is locally equivalent to a flat linear inclusion. Local flatness excludes the pathological behavior exhibited by wild counterexamples.

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Arthur Moritz SchoenfliesRelated: Higher-dimensional versions expose limits of the planar theorem associated with his name.