Schoenflies' theorem

The in 1908 by Arthur Moritz Schoenflies proven schoenflies problem is an essential link between the topology and the combinatorial problem of dyeing cards ( four color theorem ). It clearly states: If you paint a closed curve (without crossovers) on a rubber blanket, then you can warp the blanket so that the curve becomes a circle.

sentence

Let it be a closed Jordan curve and denote the unit circle . Then every homeomorphism can be continued into a homeomorphism . ${\ displaystyle K \ subset \ mathbb {R} ^ {2}}$ ${\ displaystyle S ^ {1} \ subset \ mathbb {R} ^ {2}}$ ${\ displaystyle h \ colon K \ to S ^ {1}}$ ${\ displaystyle H \ colon \ mathbb {R} ^ {2} \ to \ mathbb {R} ^ {2}}$

Higher dimensions

The direct generalization of Schoenflies' theorem to higher dimensions does not apply, since Alexander's sphere (see and weblink) offers a counterexample in three dimensions .

On the other hand, Morton Brown generalized the theorem as follows: If a -dimensional sphere is embedded locally and flat in a -dimensional sphere , then the pair is homeomorphic to , with the equator being the -sphere. (This is called an embedding locally flat when embedding is based on with matches.) ${\ displaystyle (n-1)}$ ${\ displaystyle S}$ ${\ displaystyle n}$ ${\ displaystyle S ^ {n}}$ ${\ displaystyle (S ^ {n}, S)}$ ${\ displaystyle (S ^ {n}, S ^ {n-1})}$ ${\ displaystyle S ^ {n-1}}$ ${\ displaystyle n}$ ${\ displaystyle i: S ^ {n-1} \ rightarrow S ^ {n}}$ ${\ displaystyle S ^ {n-1} \ times \ left [0,1 \ right] \ rightarrow S ^ {n}}$ ${\ displaystyle S ^ {n-1} \ times \ left \ {0 \ right \} = S ^ {n-1}}$ ${\ displaystyle i}$

This is especially true for differentiable embedded spheres, where the result is known as Mazur's theorem.

Inference

The schoenflies problem pulls directly the Jordan curve set according to: The two disjoint regions , in which is divided, are straight (the limited region) and (the unrestricted area). ${\ displaystyle \ mathbb {R} ^ {2} \ setminus K}$ ${\ displaystyle H ^ {- 1} (\ {x \ in \ mathbb {R} ^ {2}: \ | x \ | _ {2} <1 \})}$ ${\ displaystyle H ^ {- 1} (\ {x \ in \ mathbb {R} ^ {2}: \ | x \ | _ {2}> 1 \})}$

literature

Morton Brown: A proof of the generalized Schoenflies theorem. In: Bulletin of the American Mathematical Society , 66, 1960, ISSN 0002-9904 , pp. 74–76, ams.org (PDF; 280 kB)
Charles O. Christenson, William L. Voxman: Aspects of topology . Marcel Dekker, New York [u. a.] 1977, ISBN 0-8247-6331-9 .
Egbert Harzheim : Introduction to combinatorial topology (= mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X ( MR0533264 ).

Web links

Eric Weisstein: Schoenflies Theorem . In: MathWorld (English).
Alexander's "horned sphere" in the English language Wikipedia

Individual evidence

^ Charles O. Christenson, William L. Voxman: Aspects of topology . Marcel Dekker, New York [u. a.] 1977, ISBN 0-8247-6331-9 , pp. 144 .
↑ Egbert Harzheim : Introduction to Combinatorial Topology (= Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X , p. 150 ( MR0533264 ).

[1] Charles O. Christenson, William L. Voxman: Aspects of topology . Marcel Dekker, New York [u. a.] 1977, ISBN 0-8247-6331-9 , pp. 144 .

[2] Egbert Harzheim : Introduction to Combinatorial Topology (= Mathematics. Introductions to the subject matter and results of its sub-areas and related sciences ). Scientific Book Society, Darmstadt 1978, ISBN 3-534-07016-X , p. 150 ( MR0533264 ).