Schoenflies problem
In
Original formulation
The original formulation of the Schoenflies problem states that not only does every
An alternative statement is that if is a simple closed curve, then there is a homeomorphism such that 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
Such a theorem is valid only in two dimensions. In three dimensions there are
Proofs of the Jordan–Schoenflies theorem
For smooth or polygonal curves, the
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.[3]
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.[4]
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.
Continuous curve
The Jordan-Schoenflies theorem for continuous curves can be proved using
The continuous case can also be deduced from the polygonal case by approximating the continuous curve by a polygon.
Given the Jordan curve theorem, the Jordan-Schoenflies theorem can be proved as follows.[9]
- The first step is to show that a dense set of points on the curve are accessible from the inside of the curve, i.e. they are at the end of a line segment lying entirely in the interior of the curve. In fact, a given point on the curve is arbitrarily close to some point in the interior and there is a smallest closed disk about that point which intersects the curve only on its boundary; those boundary points are close to the original point on the curve and by construction are accessible.
- The second step is to prove that given finitely many accessible points Ai on the curve connected to line segments AiBi in its interior, there are disjoint polygonal curves in the interior with vertices on each of the line segments such that their distance to the original curve is arbitrarily small. This requires hexagonal tessellation; or the standard brickworktiling by rectangles or squares with common or stretch bonds. It suffices to construct a polygonal path so that its distance to the Jordan curve is arbitrarily small. Orient the tessellation such no side of a tiles is parallel to any AiBi. The size of the tiles can be taken arbitrarily small. Take the union of all the closed tiles containing at least one point of the Jordan curve. Its boundary is made up of disjoint polygonal curves. If the size of the tiles is sufficiently small, the endpoints Bi will lie in the interior of exactly one of the polygonal boundary curves. Its distance to the Jordan curve is less than twice the diameter of the tiles, so is arbitrarily small.
- The third step is to prove that any homeomorphism f between the curve and a given triangle can be extended to a homeomorphism between the closures of their interiors. In fact take a sequence ε1, ε2, ε3, ... decreasing to zero. Choose finitely many points Ai on the Jordan curve Γ with successive points less than ε1 apart. Make the construction of the second step with tiles of diameter less than ε1 and take Ci to be the points on the polygonal curve Γ1 intersecting AiBi. Take the points f(Ai) on the triangle. Fix an origin in the triangle Δ and scale the triangle to get a smaller one Δ1 at a distance less than ε1 from the original triangle. Let Di be the points at the intersection of the radius through f(Ai) and the smaller triangle. There is a piecewise linear homeomorphism F1 of the polygonal curve onto the smaller triangle carrying Ci onto Di. By the Jordan-Schoenflies theorem it extends to a homeomorphism F1 between the closure of their interiors. Now carry out the same process for ε2 with a new set of points on the Jordan curve. This will produce a second polygonal path Γ2 between Γ1 and Γ. There is likewise a second triangle Δ2 between Δ1 and Δ. The line segments for the accessible points on Γ divide the polygonal region between Γ2 and Γ1 into a union of polygonal regions; similarly for radii for the corresponding points on Δ divides the region between Δ2 and Δ1 into a union of polygonal regions. The homeomorphism F1 can be extended to homeomorphisms between the different polygons, agreeing on common edges (closed intervals on line segments or radii). By the polygonal Jordan-Schoenflies theorem, each of these homeomorphisms extends to the interior of the polygon. Together they yield a homeomorphism F2 of the closure of the interior of Γ2 onto the closure of the interior of Δ2; F2 extends F1. Continuing in this way produces polygonal curves Γn and triangles Δn with a homomeomorphism Fn between the closures of their interiors; Fn extends Fn – 1. The regions inside the Γn increase to the region inside Γ; and the triangles Δn increase to Δ. The homeomorphisms Fn patch together to give a homeomorphism F from the interior of Γ onto the interior of Δ. By construction it has limit f on the boundary curves Γ and Δ. Hence F is the required homeomorphism.
- The fourth step is to prove that any homeomorphism between Jordan curves can be extended to a homeomorphism between the closures of their interiors. By the result of the third step, it is sufficient to show that any homeomorphism of the boundary of a triangle extends to a homeomorphism of the closure of its interior. This is a consequence of the Alexander trick. (The Alexander trick also establishes a homeomorphism between the solid triangle and the closed disk: the homeomorphism is just the natural radial extension of the projection of the triangle onto its circumcircle with respect to its circumcentre.)
- The final step is to prove that given two Jordan curves there is a homeomorphism of the plane of compact support carrying one curve onto the other. In fact each Jordan curve lies inside the same large circle and in the interior of each large circle there are radii joining two diagonally opposite points to the curve. Each configuration divide the plane into the exterior of the large circle, the interior of the Jordan curve and the region between the two into two bounded regions bounded by Jordan curves (formed of two radii, a semicircle, and one of the halves of the Jordan curve). Take the identity homeomorphism of the large circle; piecewise linear homeomorphisms between the two pairs of radii; and a homeomorphism between the two pairs of halves of the Jordan curves given by a linear reparametrization. The 4 homeomorphisms patch together on the boundary arcs to yield a homeomorphism of the plane given by the identity off the large circle and carrying one Jordan curve onto the other.
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.[10] (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 f of the unit circle can be extended to a diffeomorphism F of the unit disk by the
where ψ is a smooth function with values in [0,1], equal to 0 near 0 and 1 near 1, and f(eiθ) = eig(θ), with g(θ + 2π) = g(θ) + 2π. 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,[11] 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 Hirsch (1994). 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.
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 αt 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 αs 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
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.
Notes
- ^ See:
- ^ Katok & Climenhaga 2008
- ^ See:
- ^ Moise 1977, pp. 26–29
- ^ Bing 1983, p. 29
- ^ See:
- ^ See:
- ^ See:
- Bing 1983
- Katok & Climenhaga 2008, Lecture 36
- ^ Bing & 1983, pp. 29–32
- ^ See:
- ^ See:
- Hirsch 1994, p. 182, Theorem 1.9
- Shastri 2011, p. 173, Theorem 6.4.3
- ^ See:
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