Mazur's theorem (embeddings)

In mathematics , Mazur's theorem is the higher-dimensional generalization of the Jordanian curve theorem , which however only applies to differentiable embeddings. It is proven using Morse theory .

The ellipse on the spherical surface divides it into two parts, which in turn are both homeomorphic to a circle.

statement

A -sphere differentiable embedded in the -dimensional sphere divides the into two connected components , both of which are homeomorphic to the -dimensional solid sphere . ${\displaystyle n}$ ${\displaystyle S^{n}}$ ${\displaystyle (n-1)}$ ${\displaystyle S^{n}}$ ${\displaystyle n}$

Dimension 2

The Jordanian curve set is obtained in 2 dimensions, at least for differentiable Jordan curves. To do this, map the plane together with a differentiable Jordan curve contained therein by means of reverse stereographic projection onto the spherical surface, then only one point, such as a north pole, is added. The above sentence then guarantees the decomposition into two connected components, for example as in the adjacent illustration, of which the unrestricted one in the plane is the one that contains the North Pole.

Counterexample

For topological (non-differentiable) embeddings, the theorem in dimensions no longer applies , a counterexample is Alexander's sphere . In any case, the Jordan-Brouwer decomposition theorem applies , according to which the complement of an embedded sphere always consists of two connected components. ${\displaystyle n\geq 3}$

literature

• Barry Mazur : On embeddings of spheres. In: Bulletin of the American Mathematical Society . Volume 65, Number 2, 1959, pp. 59-65, ( online ).
• Maxwell HA Newman : On the division of Euclidean n-sphere by topological (n-1) -spheres. In: Proceedings of the Royal Society of London . Series A: Mathematical and Physical Sciences. Volume 257, Number 1288, Aug. 23, 1960, pp. 1-12, ( JSTOR 2413794 ).

Web links

• Mazur's Theorem (Wolfram MathWorld)