Schoenflies problem

In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.
Original formulation.
The original formulation of the Schoenflies problem 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 in the plane.
An alternative statement is that if formula_1 is a simple closed curve, then there is a homeomorphism formula_2 such that formula_3 is the unit circle in the plane. Elementary proofs can be found in , , and . The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity map off some compact set; the case of a continuous curve is then deduced by approximating by polygons. The theorem is also an immediate consequence of Carathéodory's extension theorem for conformal mappings, as discussed in .
If the curve is smooth then the homeomorphism can be chosen to be a diffeomorphism. Proofs in this case rely on techniques from differential topology. Although direct proofs are possible (starting for example from the polygonal case), existence of the diffeomorphism can also be deduced by using the smooth Riemann mapping theorem for the interior and exterior of the curve in combination with the Alexander trick for diffeomorphisms of the circle and a result on smooth isotopy from differential topology.
Such a theorem is valid only 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.
Proofs of the Jordan–Schoenflies theorem.
For smooth or polygonal curves, the Jordan curve theorem can be proved in a straightforward way. Indeed the curve has a tubular neighbourhood, defined in the smooth case by the field of unit normal vectors to the curve or in the polygonal case by points at a distance of less than ε from the curve.
In a neighbourhood of a differentiable point on the curve, there is a coordinate change in which the curve becomes the diameter of an open disk. Taking a point not on the curve, a straight line aimed at the curve starting at the point will eventually meet the tubular neighborhood; the path can be continued next to the curve until it meets the disk. It will meet it on one side or the other. This proves that the complement of the curve has at most two connected components. On the other hand using the Cauchy integral formula for the winding number, it can be seen that the winding number is constant on connected components of the complement of the curve, is zero near infinity and increases by 1 when crossing the curve. Hence the curve has exactly two components, its interior and the unbounded component. The same argument works for a piecewise differentiable Jordan curve.
Polygonal curve.
Given a simple closed polygonal curve in the plane, the piecewise linear Jordan–Schoenflies theorem states that there is a piecewise linear homeomorphism of the plane, with compact support, carrying the polygon onto a triangle and taking the interior and exterior of one onto the interior and exterior of the other.
The interior of the polygon can be triangulated by small triangles, so that the edges of the polygon form edges of some of the small triangles. Piecewise linear homeomorphisms can be made up from special homeomorphisms obtained by removing a diamond from the plane and taking a piecewise affine map, fixing the edges of the diamond, but moving one diagonal into a V shape. Compositions of homeomorphisms of this kind give rise to piecewise linear homeomorphisms of compact support; they fix the outside of a polygon and act in an affine way on a triangulation of the interior. A simple inductive argument shows that it is always possible to remove a "free" triangle—one for which the intersection with the boundary is a connected set made up of one or two edges—leaving a simple closed Jordan polygon. The special homeomorphisms described above or their inverses provide piecewise linear homeomorphisms which carry the interior of the larger polygon onto the polygon with the free triangle removed. Iterating this process it follows that there is a piecewise linear homeomorphism of compact support carrying the original polygon onto a triangle.
Because the homeomorphism is obtained by composing finite many homeomorphisms of the plane of compact support, it follows that the piecewise linear homeomorphism in the statement of the piecewise linear Jordan-Schoenflies theorem has compact support.
As a corollary, it follows that any homeomorphism between simple closed polygonal curves extends to a homeomorphism between their interiors. For each polygon there is a homeomorphism of a given triangle onto the closure of their interior. The three homeomorphisms yield a single homeomorphism of the boundary of the triangle. By the Alexander trick this homeomorphism can be extended to a homeomorphism of closure of interior of the triangle. Reversing this process this homeomorphism yields a homeomorphism between the closures of the interiors of the polygonal curves.
Continuous curve.
The Jordan-Schoenflies theorem for continuous curves can be proved using Carathéodory's theorem on conformal mapping. It states that the Riemann mapping between the interior of a simple Jordan curve and the open unit disk extends continuously to a homeomorphism between their closures, mapping the Jordan curve homeomorphically onto the unit circle. To prove the theorem, Carathéodory's theorem can be applied to the two regions on the Riemann sphere defined by the Jordan curve. This will result in homeomorphisms between their closures and the closed disks |"z"| ≤ 1 and |"z"| ≥ 1. The homeomorphisms from the Jordan curve to
the circle will differ by a homeomorphism of the circle which can be extended to the unit disk (or its complement) by the Alexander trick. Composition with this homeomorphism will yield a pair of homeomorphisms which match on the Jordan curve and therefore define a homeomorphism of the Riemann sphere carrying the Jordan curve onto the unit circle.
The continuous case can also be deduced from the polygonal case by approximating the continuous curve by a polygon. The Jordan curve theorem is first deduced by this method. The Jordan curve is given by a continuous function on the unit circle. It and the inverse function from its image back to the unit circle are uniformly continuous. So dividing the circle up into small enough intervals, there are points on the curve such that the line segments joining adjacent points lie close to the curve, say by ε. Together these line segments form a polygonal curve. If it has self-intersections, these must also create polygonal loops. Erasing these loops, results in a polygonal curve without self-intersections which still lies close to the curve; some of its vertices might not lie on the curve, but they all lie within a neighbourhood of the curve. The polygonal curve divides the plane into two regions, one bounded region "U" and one unbounded region "V". Both "U" and "V" ∪ ∞ are continuous images of the closed unit disk. Since the original curve is contained within a small neighbourhood of the polygonal curve, the union of the images of slightly smaller concentric open disks entirely misses the original curve and their union excludes a small neighbourhood of the curve. One of the images is a bounded open set consisting of points around which the curve has winding number one; the other is an unbounded open set consisting of points of winding number zero. Repeating for a sequence of values of ε tending to 0, leads to a union of open path-connected bounded sets of points of winding number one and a union of open path-connected unbounded sets of winding number zero. By construction these two disjoint open path-connected sets fill out the complement of the curve in the plane.
Given the Jordan curve theorem, the Jordan-Schoenflies theorem can be proved as follows.
Smooth curve.
Proofs in the smooth case depend on finding a diffeomorphism between the interior/exterior of the curve and the closed unit disk (or its complement in the extended plane). This can be solved for example by using the smooth Riemann mapping theorem, for which a number of direct methods are available, for example through the Dirichlet problem on the curve or Bergman kernels. (Such diffeomorphisms will be holomorphic on the interior and exterior of the curve; more general diffeomorphisms can be constructed more easily using vector fields and flows.) Regarding the smooth curve as lying inside the extended plane or 2-sphere, these analytic methods produce smooth maps up to the boundary between the closure of the interior/exterior of the smooth curve and those of the unit circle. The two identifications of the smooth curve and the unit circle will 
differ by a diffeomorphism of the unit circle. On the other hand a diffeomorphism of the unit circle can be extended to a diffeomorphism of the unit disk by the Alexander extension:
where is a smooth function with values in [0,1], equal to 0 near 0 and 1 near 1, and , with . Composing one of the diffeomorphisms with the Alexander extension allows the two diffeomorphisms to be patched together to give a homeomorphism of the 2-sphere which restricts to a diffeomorphism on the closed unit disk and the closures of its complement which it carries onto the interior and exterior of the original smooth curve. By the "isotopy theorem" in differential topology, the homeomorphism can be adjusted to a diffeomorphism on the whole 2-sphere without changing it on the unit circle. This diffeomorphism then provides the smooth solution to the Schoenflies problem.
The Jordan-Schoenflies theorem can be deduced using differential topology. In fact it is an immediate consequence of the classification up to diffeomorphism of smooth oriented 2-manifolds with boundary, as described in . Indeed the smooth curve divides the 2-sphere into two parts. By the classification each is diffeomorphic to the unit disk and—taking into account the isotopy theorem—they are glued together by a diffeomorphism of the boundary. By the Alexander trick, such a diffeomorphism extends to the disk itself. Thus there is a diffeomorphism of the 2-sphere carrying the smooth curve onto the unit circle.
On the other hand the diffeomorphism can also be constructed directly using the Jordan-Schoenflies theorem for polygons and elementary methods from differential topology, namely flows defined by vector fields. When the Jordan curve is smooth (parametrized by arc length) the unit normal vectors give a non-vanishing vector field "X"0 in a tubular neighbourhood "U"0 of the curve. Take a polygonal curve in the interior of the curve close to the boundary and transverse to the curve (at the vertices the vector field should be strictly within the angle formed by the edges). By the piecewise linear Jordan–Schoenflies theorem, there is a piecewise linear homeomorphism, affine on an appropriate triangulation of the interior of the polygon, taking the polygon onto a triangle. Take an interior point "P" in one of the small triangles of the triangulation. It corresponds to a point "Q" in the image triangle. There is a radial vector field on the image triangle, formed of straight lines pointing towards "Q". This gives a series of lines in the small triangles making up the polygon. Each defines a vector field "X""i" on a neighbourhood "U""i" of the closure of the triangle. Each vector field is transverse to the sides, provided that "Q" is chosen in "general position" so that it is not collinear with any of the finitely many edges in the triangulation. Translating if necessary, it can be assumed that "P" and "Q" are at the origin 0. On the triangle containing "P" the vector field can be taken to be the standard radial vector field. Similarly the same procedure can be applied to the outside of the smooth curve, after applying Möbius transformation to map it into the finite part of the plane and ∞ to 0. In this case the neighbourhoods "U""i" of the triangles have negative indices. Take the vector fields "X""i" with a negative sign, pointing away from the point at infinity. Together "U"0 and the "U""i"'s with "i" ≠ 0 form an open cover of the 2-sphere. Take a smooth partition of unity ψ"i" subordinate to the cover "U""i" and set
"X" is a smooth vector field on the two sphere vanishing only at 0 and ∞. It has index 1 at 0 and -1 at ∞. Near 0 the vector field equals the radial vector field pointing towards 0. If α"t" is the smooth flow defined by "X", the point 0 is an attracting point and ∞ a repelling point. As "t" tends to +∞, the flow send points to 0; while as "t" tends to –∞ points are sent to ∞. Replacing "X" by "f"⋅"X" with "f" a smooth positive function, changes the parametrization of the integral curves of "X", but not the integral curves themselves. For an appropriate choice of "f" equal to 1 outside a small annulus near 0, the integral curves starting at points of the smooth curve will all reach smaller circle bounding the annulus at the same time "s". The diffeomorphism α"s" therefore carries the smooth curve onto this small circle. A scaling transformation, fixing 0 and ∞, then carries the small circle onto the unit circle. Composing these diffeomorphisms gives a diffeomorphism carrying the smooth curve onto the unit circle.
Generalizations.
There does exist a higher-dimensional generalization due to and independently with , which is also called the generalized Schoenflies 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. Both the Brown and Mazur proofs are considered "elementary" and use inductive arguments.
The Schoenflies 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. For "n" ≥ 5 the question in the smooth category has an affirmative answer, and follows from the h-cobordism theorem.