# Embedding (simple definition)

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## 1 Definition

A smooth embedding of a smooth compact manifold $N$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}N$ into a smooth manifold is a smooth injective map $f:N\to M$$f:N\to M$ such that $df$$df$ is a monomorphism at each point. (See an equivalent alternative definition which works for non-compact manifolds and involves immersions. A smooth immersion is a smooth map $f:N\to M$$f:N\to M$ such that $df$$df$ is a monomorphism at each point. See an equivalent alternative definition.)

A map $f:N\to\Rr^m$$f:N\to\Rr^m$ of a polyhedron $N$$N$ is piecewise-linear (PL) if it is linear on each simplex of some smooth triangulation of $N$$N$. A PL embedding of a compact polyhedron $N$$N$ into $\Rr^m$$\Rr^m$ is a PL injective map $f:N\to\Rr^m$$f:N\to\Rr^m$.

A topological embedding of a compact subset $N\subset\Rr^n$$N\subset\Rr^n$ into $\Rr^m$$\Rr^m$ is a continuous injective map $f:N\to\Rr^m$$f:N\to\Rr^m$.

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