Locally flat embedding
In mathematics , locally shallow embedding is a term from the topology of manifolds . In many cases, locally shallow embeddings are easier to classify. For various classical theorems, such as Schoenflies' theorem , local flatness is the most general possible requirement.
definition
An embedding
between (topological) manifolds is called locally flat if there are maps and and a homeomorphism for every point
where the -dimensional unit sphere denotes.
Examples
- Every smooth embedding between differentiable manifolds is locally flat. In this case, the cards used in the definition can even be selected to be differentiable.
- The Alexander sphere is a 2-sphere embedded in the , which is not locally flat.
Applications
The classic schoenflies problem stating that it at any closed Jordan curve a homeomorphism is the mapping of the unit circle. This sentence cannot be transferred directly to higher dimensions, e.g. a. because the Alexander Sphere provides a counterexample. However, Schoenflies' theorem can be generalized for locally shallow embeddings.
Brown's theorem : If an (n-1) -dimensional sphere S is embedded locally flat in the n-dimensional Euclidean space , then the pair is homeomorphic to , where the (n-1) -dimensional unit sphere denotes.
Brown's theorem also applies analogously to locally flat embeddings of codimension ≥ 3 (where it was proven by Stallings , which is why the general formulation is also known as the Brown-Stallings theorem), while in codimension 2 there is the phenomenon of knotting .
literature
- Barry Mazur: On embeddings of spheres. Bulletin of the American Mathematical Society, Vol. 65 (1959), no. 2, pp. 59-65. online (pdf)
- Morton Brown: Locally flat imbeddings of topological manifolds. Annals of Mathematics, Second series, Vol. 75 (1962), pp. 331-341. online (pdf)
- LV Keldysh, AV Chernavskii: Topological imbeddings in Euclidean space . Proc. Steklov Inst. Math. 81 (1968) Trudy Mat. Inst. Steklov. online (pdf)
Web links
- Isotopy (Encyclopedia of Mathematics)