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.

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 ${\displaystyle C\subset \mathbb {R} ^{2}}$ is a simple closed curve, then there is a homeomorphism ${\displaystyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}$ such that ${\displaystyle f(C)}$ is the unit circle in the plane. Elementary proofs can be found in Newman (1939), Cairns (1951), Moise (1977) and Thomassen (1992). 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 Pommerenke (1992, p. 25).

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.

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 separates the plane into exactly two components, its "interior" and its "exterior", the latter being unbounded. The same argument works for a piecewise differentiable Jordan 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.