Embedding (simple definition)
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A ''smooth embedding'' of a smooth compact manifold $N$ into a smooth manifold is a smooth injective map $f:N\to M$ such that | A ''smooth embedding'' of a smooth compact manifold $N$ into a smooth manifold is a smooth injective map $f:N\to M$ such that | ||
$df$ is a monomorphism at each point. | $df$ is a monomorphism at each point. | ||
− | (See an [[Embedding#Definition|alternative definition]] which works for non-compact manifolds and involves immersions. | + | (See an [[Embedding#Definition|equivalent alternative definition]] which works for non-compact manifolds and involves immersions. |
A ''smooth immersion'' is a smooth map $f:N\to M$ such that $df$ is a monomorphism at each point. | A ''smooth immersion'' is a smooth map $f:N\to M$ such that $df$ is a monomorphism at each point. | ||
− | See an [[Immersion#Definition|alternative definition]].) | + | See an [[Immersion#Definition|equivalent alternative definition]].) |
A map $f:N\to\Rr^m$ of a polyhedron $N$ is ''piecewise-linear (PL)'' if it is linear on each simplex of some smooth triangulation of $N$. | A map $f:N\to\Rr^m$ of a polyhedron $N$ is ''piecewise-linear (PL)'' if it is linear on each simplex of some smooth triangulation of $N$. | ||
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A ''topological embedding'' of a compact subset $N\subset\Rr^n$ into $\Rr^m$ is a continuous injective map $f:N\to\Rr^m$. | A ''topological embedding'' of a compact subset $N\subset\Rr^n$ into $\Rr^m$ is a continuous injective map $f:N\to\Rr^m$. | ||
+ | [[Embeddings_in_Euclidean_space:_an_introduction_to_their_classification|Classification]] of embeddings up to [[Isotopy|isotopy]] is a classical problem in topology, see \cite{Skopenkov2016c}. | ||
</wikitex> | </wikitex> | ||
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[[Category:Definitions]] | [[Category:Definitions]] | ||
+ | [[Category:Embeddings of manifolds]] |
Latest revision as of 09:42, 16 April 2018
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Definition
A smooth embedding of a smooth compact manifold into a smooth manifold is a smooth injective map such that is a monomorphism at each point. (See an equivalent alternative definition which works for non-compact manifolds and involves immersions. A smooth immersion is a smooth map such that is a monomorphism at each point. See an equivalent alternative definition.)
A map of a polyhedron is piecewise-linear (PL) if it is linear on each simplex of some smooth triangulation of . A PL embedding of a compact polyhedron into is a PL injective map .
A topological embedding of a compact subset into is a continuous injective map .
Classification of embeddings up to isotopy is a classical problem in topology, see [Skopenkov2016c].
2 References
- [Skopenkov2016c] A. Skopenkov, Embeddings in Euclidean space: an introduction to their classification, to appear to Bull. Man. Atl.