Schoenflies problem

Elementary proofs can be found in Newman (1939), Cairns (1951), Moise (1977) and Thomassen (1992).

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.

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.

[2] 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.

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.

Iterating this process it follows that there is a piecewise linear homeomorphism of compact support carrying the original polygon onto a triangle.

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.

It and the inverse function from its image back to the unit circle are uniformly continuous.

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.

[9] 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.

By the isotopy theorem in differential topology,[11] the homeomorphism can be adjusted to a diffeomorphism on the whole 2-sphere without changing it on the unit circle.

In fact it is an immediate consequence of the classification up to diffeomorphism of smooth oriented 2-manifolds with boundary, as described in Hirsch (1994).

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.

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 Xi on a neighbourhood Ui 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.

Both the Brown and Mazur proofs are considered "elementary" and use inductive arguments.

For n ≥ 5 the question in the smooth category has an affirmative answer, and follows from the h-cobordism theorem.