Schönflies problem
Encyclopedia
In mathematics
, the Schoenflies problem or Schoenflies theorem, of geometric topology
is a sharpening of the Jordan curve theorem
by Arthur Schoenflies.
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
is a simple closed curve, then there is a homeomorphism 
such that 
is the unit circle
in the plane.
Such a theorem is only valid in two dimensions. In three dimensions there are counterexample
s 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.
and independently Barry Mazur
with Marston Morse
, 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, Sn−1), where Sn−1 is the equator of the n-sphere. Brown and Mazur received the Veblen Prize for their contributions.
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 has an affirmative answer, and follows from the h-cobordism
theorem.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Schoenflies problem or Schoenflies theorem, of geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
is a sharpening of the Jordan curve theorem
Jordan curve theorem
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
by Arthur Schoenflies.
Original formulation
It states that not only does every simple closed curve in the planePlane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
separate the plane into two regions, one (the "inside") bounded
Bounded set
In mathematical analysis and related areas of mathematics, a set is called bounded, if it is, in a certain sense, of finite size. Conversely, a set which is not bounded is called unbounded...
and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
in the plane.
An alternative statement is that if
is a simple closed curve, then there is a homeomorphism such that is the unit circleCircle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
in the plane.
Such a theorem is only valid in two dimensions. In three dimensions there are counterexample
Counterexample
In logic, and especially in its applications to mathematics and philosophy, a counterexample is an exception to a proposed general rule. For example, consider the proposition "all students are lazy"....
s 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.
Generalizations
There does exist a higher-dimensional generalization due to Morton BrownMorton Brown
Morton Brown is an American mathematician, who specializes in geometric topology.In 1958 Brown earned his PhD from the University of Wisconsin at Madison under R. H. Bing. From 1960 to 1962 he was at the Institute for Advanced Study...
and independently Barry Mazur
Barry Mazur
-Life:Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. Regardless, he was accepted for graduate school and received his Ph.D. from Princeton University in 1959,...
with Marston Morse
Marston Morse
Harold Calvin Marston Morse was an American mathematician best known for his work on the calculus of variations in the large, a subject where he introduced the technique of differential topology now known as Morse theory...
, which is also called the generalized Schoenflies theorem. It states that, if an (n − 1)-dimensional sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
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, Sn−1), where Sn−1 is the equator of the n-sphere. Brown and Mazur received the Veblen Prize for their contributions.
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
Mazur manifold
In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth 4-dimensional manifold which is not diffeomorphic to the standard 4-ball...
. For n ≥ 5 the question has an affirmative answer, and follows from the h-cobordism
H-cobordism
A cobordism W between M and N is an h-cobordism if the inclusion mapsare homotopy equivalences...
theorem.