Embedding

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1 Definition

We work in a fixed category CAT of topological, piecewise linear, $C^r$${{Authors|Ulrich Koschorke}} == 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 [[2-manifolds|smooth surfaces]] can be immersed into $\R^3$; but [[2-manifolds#Non-orientable_surfaces|non-orientable 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 more modern expositions see also \cite{Adachi1993|p. 67ff} and \cite{Prasolov2007|22.1}. {{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. The difficulty of this task depends heavily on the choices of $M$ and $N$ and especially their dimensions: for more information please see the page on [[Embeddings in Euclidean space: an introduction to their classification|high codimension embeddings]]. 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 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

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$M$ 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

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$M$ 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

Tex syntax error

$M$ there exists a $C^{\infty}$$C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$$M \hookrightarrow \R^{2m}$.

Remark 2.4. For more modern expositions see also [Adachi1993, p. 67ff] and [Prasolov2007, 22.1].

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

Tex syntax error

$M$ 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

Tex syntax error

$M$ and $N$$N$ this may turn out to be a very difficult task. E.g. in the theory of knots (or links) where

Tex syntax error

$M$ 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.

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
• [Prasolov2007] V. V. Prasolov, Elements of homology theory, American Mathematical Society, 2007. MR2313004 (2008d:55001) Zbl 1120.55001
• [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

$-sphere (or a finite union of 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

Tex syntax error

$M$ 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

Tex syntax error

$M$ 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

Tex syntax error

$M$ there exists a $C^{\infty}$$C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$$M \hookrightarrow \R^{2m}$.

Remark 2.4. For more modern expositions see also [Adachi1993, p. 67ff] and [Prasolov2007, 22.1].

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

Tex syntax error

$M$ 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

Tex syntax error

$M$ and $N$$N$ this may turn out to be a very difficult task. E.g. in the theory of knots (or links) where

Tex syntax error

$M$ 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.

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
• [Prasolov2007] V. V. Prasolov, Elements of homology theory, American Mathematical Society, 2007. MR2313004 (2008d:55001) Zbl 1120.55001
• [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

$-spheres), and $N = \R^{3}$ 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. == 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

Tex syntax error

$M$ 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

Tex syntax error

$M$ 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

Tex syntax error

$M$ there exists a $C^{\infty}$$C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$$M \hookrightarrow \R^{2m}$.

Remark 2.4. For more modern expositions see also [Adachi1993, p. 67ff] and [Prasolov2007, 22.1].

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

Tex syntax error

$M$ 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

Tex syntax error

$M$ and $N$$N$ this may turn out to be a very difficult task. E.g. in the theory of knots (or links) where

Tex syntax error

$M$ 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.

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
• [Prasolov2007] V. V. Prasolov, Elements of homology theory, American Mathematical Society, 2007. MR2313004 (2008d:55001) Zbl 1120.55001
• [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