# Embedding

 An earlier version of this page was published in the Definitions section of the Bulletin of the Manifold Atlas: screen, print. You may view the version used for publication as of 12:03, 16 May 2013 and the changes since publication.

## 1 Definition

We work in a fixed category CAT of topological, piecewise linear, $C^r$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}C^r$-differentiable $(1 \leq r \leq \infty )$$(1 \leq r \leq \infty )$ or real analytic manifolds (second countable, Hausdorff, without boundary) and maps between them.

Let $f : M^m \rightarrow N^n$$f : M^m \rightarrow N^n$ be such a map between manifolds of the indicated dimensions $1 \leq m < n$$1 \leq m < n$.

Definition 1.1. We call $f$$f$ an embedding (and we write $f : M \hookrightarrow N$$f : M \hookrightarrow N$) if $f$$f$ is an immersion which maps $M$$M$ homeomorphically onto its image.

It follows that an embedding cannot have selfintersections. But even an injective immersion need not be an embedding; e. g. the figure six 6 is the image of a smooth immersion but not of an embedding. Note that in the topological and piecewise linear categories, CAT = TOP or PL, our definition yields locally flat embeddings. In these categories there are other concepts of embeddings - e.g. wild embeddings - which are not locally flat: the condition of local flatness is implied by our definition of immersion. Embeddings (and immersions) into familiar target manifolds such as $\R^n$$\R^n$ may help to visualize abstractly defined manifolds. E. g. all smooth surfaces can be immersed into $\R^3$$\R^3$; but non-orientable surfaces (such as the projective plane and the Klein bottle) allow no embeddings into $\R^3$$\R^3$.

## 2 Existence of embeddings

Theorem 2.1 [Penrose&Whitehead&Zeeman1961]. For every compact $m$$m$--dimensional PL-manifold $M$$M$ there exists a PL--embedding $M \hookrightarrow \R^{2m}$$M \hookrightarrow \R^{2m}$.

Remark 2.2. For a good exposition of Theorem 2.1 see also [Rourke&Sanderson1972a, p. 63].

Theorem 2.3 [Whitney1944]. For every closed m--dimensional $C^{\infty}$$C^{\infty}$--manifold $M$$M$ there exists a $C^{\infty}$$C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$$M \hookrightarrow \R^{2m}$.

Similar existence results for embeddings $M^m \hookrightarrow \R^N$$M^m \hookrightarrow \R^N$ are valid also in the categories of real analytic maps and of isometrics (Nash) when $N \gg 2m$$N \gg 2m$ is sufficiently high.

## 3 Classification

In order to get a survey of all essentially distinct´´ embeddings $f : M \hookrightarrow N$$f : M \hookrightarrow N$ it is meaningful to introduce equivalence relations such as (ambient) isotopy, concordance, bordism etc., and to aim at classifying embeddings accordingly. The difficulty of this task depends heavily on the choices of $M$$M$ and $N$$N$ and especially their dimensions: for more information please see the page on high codimension embeddings. Already for the most basic choices of $M$$M$ and $N$$N$ this may turn out to be a very difficult task. E.g. in the theory of knots (or links) where $M$$M$ is a $1$$1$-sphere (or a finite union of $1$$1$-spheres), and $N = \R^{3}$$N = \R^{3}$ the multitude of possible knotting and linking phenomena is just overwhelming. Even classifying links up to the very crude equivalence relation `link homotopy´ is very far from having been achieved yet.