Embedding

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1 Definition

We work in a fixed category CAT of topological, piecewise linear, $C^r$${{Authors|Ulrich Koschorke}}{{Definition reviewed}} == Definition == <wikitex>; We work in a fixed category CAT of topological, piecewise linear, $ C^r$-differentiable $(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 $ be such a map between manifolds of the indicated dimensions $ 1 \leq m < n $. {{beginthm|Definition}} We call $f$ an '''embedding''' (and we write $ f : M \hookrightarrow N $) if $f$ is an [[Immersion|immersion]] which maps $M$ homeomorphically onto its image. {{endthm}} It follows that an embedding cannot have selfintersections. But even an injective immersion need not be an embedding; e. g. the figure six 6<!-- FIXME figure 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$ may help to visualize abstractly defined manifolds. E. g. all smooth surfaces can be immersed into $\R^3$; but nonorientable surfaces (such as the projective plane and the Klein bottle) allow no embeddings into $\R^3$. </wikitex> ==Existence of embeddings== <wikitex>; {{beginthm|Theorem|\cite{Penrose&Whitehead&Zeeman1961}}}\label{thm:2.1} For every compact $m$--dimensional PL-manifold $M$ there exists a PL--embedding $ M \hookrightarrow \R^{2m}$. {{endthm}} {{beginrem|Remark}} For a good exposition of Theorem \ref{thm:2.1} see also \cite{Rourke&Sanderson1972a|p. 63}. {{endrem}} {{beginthm|Theorem|\cite{Whitney1944}}}\label{thm:2.2} For every closed m--dimensional $C^{\infty}$--manifold $M$ there exists a $C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$. {{endthm}} {{beginrem|Remark}} For a more modern exposition see also \cite{Adachi1993|p. 67ff}. {{endrem}} Similar existence results for embeddings $ M^m \hookrightarrow \R^N $ are valid also in the categories of real analytic maps and of isometrics (Nash) when $ N \gg 2m $ is sufficiently high. </wikitex> ==Classification== <wikitex>; In order to get a survey of all ``essentially distinct´´ embeddings $ 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. Already for the most basic choices of $M$ and $N$ this may turn out to be a very difficult task. E.g. in the [[Wikipedia:Knot theory|theory of knots]] (or links) where $M$ is a sphere (or a finite union of spheres) and $N = \R^n$ 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. </wikitex> == References == {{#RefList:}} == External links == * The Wikipedia page about [[Wikipedia:Embedding#Differential_topology|embeddings]] [[Category:Definitions]] [[Category:Embeddings of manifolds]]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 nonorientable 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}$.

Remark 2.4. For a more modern exposition see also [Adachi1993, p. 67ff].

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$$f : M \hookrightarrow N$ it is meaningful to introduce equivalence relations such as (ambient) isotopy, concordance, bordism etc., and to aim at classifying embeddings accordingly. Already for the most basic choices of $M$$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 sphere (or a finite union of spheres) and $N = \R^n$$N = \R^n$ 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.

4 References

• [Adachi1993] M. Adachi, Embeddings and immersions, Translated from the Japanese by Kiki Hudson. Translations of Mathematical Monographs, 124. Providence, RI: American Mathematical Society (AMS), 1993. MR1225100 (95a:57039) Zbl 0810.57001
• [Penrose&Whitehead&Zeeman1961] R. Penrose, J. Whitehead and E. Zeeman, Imbedding of manifolds in Euclidean space., Ann. of Math. 73 (1961) 613–623. MR0124909 (23 #A2218) Zbl 0113.38101
• [Rourke&Sanderson1972a] C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Springer-Verlag, 1972. MR0350744 (50 #3236) Zbl 0477.57003
• [Whitney1944] H. Whitney, The self-intersections of a smooth $n$$n$-manifold in $2n$$2n$-space, Ann. of Math. (2) 45 (1944), 220–246. MR0010274 (5,273g) Zbl 0063.08237

5 External links

• The Wikipedia page about embeddings