Embedding (simple definition)

Revision as of 10:20, 14 September 2017

Definition

A smooth embedding of a smooth compact manifold ${{Stub}} == Definition == <wikitex>; 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. (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. 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 ''PL embedding'' of a compact polyhedron $N$ into $\Rr^m$ is a PL 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 $f:N\to\Rr^m$. [[High codimension embeddings: classification|Classification]] of embeddings up to [[Isotopy|isotopy]] is a classical problem in topology, see \cite{Skopenkov2016c}. </wikitex> [[Category:Definitions]] [[Category:Embeddings of manifolds]]N$ into a smooth manifold is a smooth injective map $f:N\to M$ such that $df$ 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 $f:N\to M$ such that $df$ is a monomorphism at each point. See an 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 PL embedding of a compact polyhedron $N$ into $\Rr^m$ is a PL 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 $f:N\to\Rr^m$.

Classification of embeddings up to isotopy is a classical problem in topology, see [Skopenkov2016c].