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

${\displaystyle i\colon N\to M}$

between (topological) manifolds is called locally flat if there are maps and and a homeomorphism for every point ${\displaystyle x\in N}$ ${\displaystyle x\in U}$ ${\displaystyle i(x)\in V}$

${\displaystyle (V,i(U))\simeq (D^{m},D^{n})}$

where the -dimensional unit sphere denotes. ${\displaystyle D^{m}}$ ${\displaystyle m}$

Alexander sphere

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. ${\displaystyle \mathbb {R} ^{3}}$

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. ${\displaystyle i\colon S^{1}\to \mathbb {R} ^{2}}$ ${\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ^{2}}$ ${\displaystyle i(S^{1})}$

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. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle (R^{n},i(S))}$ ${\displaystyle (\mathbb {R} ^{n},S^{n-1})}$ ${\displaystyle S^{n-1}}$

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)