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Definition

We work in a fixed category CAT of topological, piecewise linear, $C^r$$<!--<bibitemswithcorrections/>--> <!-- {{#dpl: |namespace=Template |titlematch=%Milnor% |resultsheader=<h3 id="letter:{{{1}}}">{{{1}}}</h3> |noresultsheader=<div style="display:none"></div> |shownamespace=false |mode=userformat |listseparators=,\n* [[%PAGE%|[%TITLE%]]] ,, |includepage=* |ordermethod=title }} --> == Images == <wikitex>; {{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}} {{beginthm|Theorem}} For a good exposition of Theorem \ref{thm:2.1} see also \cite{Rourke&Sanderson1972a|p. 63}. {{endthm}} {{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]] \begin{theorem} \label{thm:1} We have $f \colon X \to Y$ \end{theorem} Reference \ref{thm:1} By Theorem {{equation|$\alpha$|1.2}} {{eqref|1.2}} {{begineq|eqtest|$\alpha$}}<label>qtest</label>{{endeq}} {{beginthm|Theorem}} Frog {{endthm}} \ref{qtest} \ref{eqtest} Here is some text leading up to an equation {{beginrem| }} $$ A = B $$ {{endrem}} Here is some more text after the equation to see how it looks. Here is some text leading up to an equation $$ A = B $$ Here is some more text after the equation to see how it looks. :{| border="1" cellpadding="2" class="wikitable" style="text-align:center" |- ! 4k !! 8 !! 12 !! 16 !! 20 !! 24 !! 28 !! 32 |- ! order bP<sub>4k</sub> | 2<sup>2</sup>.7 || 2<sup>5</sup>.31 || 2<sup>6</sup>.127|| 2<sup>9</sup>.511|| 2<sup>10</sup>.2047.691 || 2<sup>13</sup>.8191 || 2<sup>14</sup>.16384.3617 |- |} <!-- !! 36 !! 40 !! 44 !! 48 !! 52 !! 56 !! 60 !! 64 |- || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || --> :{| border="1" cellpadding="2" class="wikitable" style="text-align:center" |- ! k !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 |- ! B<sub>k</sub> | 1/6 || 1/30 || 1/42 || 1/30 || 5/66 || 691/2730 || 7/6 || 3617/510 || |- |} :{| border="1" cellpadding="2" class="wikitable" style="text-align:center" |- ! Dim n !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20 |- ! order Θ<sub>''n''</sub> | 1 || 1 || 1 || 1 || 1 || 1 || 28 || 2 || 8 || 6 || 992 || 1 || 3 || 2 || 16256 || 2 || 16 || 16 || 523264 || 24 |- !''bP''<sub>''n''+1</sub> | 1 || 1 || 1 || 1 || 1 || 1 || 28 || 1 || 2 || 1 || 992 || 1 || 1 || 1 || 8128 || 1 || 2 || 1 || 261632 || 1 |- !Θ<sub>''n''</sub>/''bP''<sub>''n''+1</sub> | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||2×2 || 6 || 1 || 1 || 3 || 2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24 |- !π<sub>''n''</sub><sup>''S''</sup>/''J'' | 1 || 2 || 1 || 1 || 1 || 2 || 1 || 2 || 2×2 || 6 || 1 || 1 || 3 || 2×2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24 |- !index | - || 2 || - || - || - || 2 || - || - || - || - || - || - || - || 2 || - || - || - || - || - || - |} [[Media:Spaceform.tiff|link text]] $$ f \colon X \to Y $$ <DPL> namespace=Main nottitlematch=Main_Page|Sandbox|Links ordermethod=lastedit order=descending minoredits=exclude author=%Diarmuid Crowley% count=10 </DPL> Just a fest $f \colon A \to B$. $\Q$ {{beginthm|a theorem}} \label{thma} {{endthm}} $\text{Spin}$ by theorem \ref{thma} <ol start="9"> <li>Amsterdam</li> <li>Rotterdam</li> <li>The Hague</li> </ol> <nowiki>[</nowiki>[[#{{anchorencode:Mess1990}}|Mess1990]]<nowiki>]</nowiki>{{#RefAdd:Mess1990}} <!-- $\sf{no serifs please}$ --> $\left( \begin{array}{ll} \alpha & \beta \ \gamma & \delta \end{array} \right)$ $f = T$ $ f : X \to Y$ $$ f : X \to Y $$ $\Ker$ $\mathscr{A}$ $\mathscr{B}$ \bf{ bold} \it{italic} \em{emphasis} </wikitex> [[Image:Foliation.png|thumb|300px|3-dimensional Reeb foliation]] == Tests == {{citeD|Ranicki1981}} {{cite|Milnor1956}} {{cite|Milnor1956|Theorem 1}} {{citeD|Milnor1956}} {{citeD|Milnor1956|Theorem 1}} {{citeD2|Milnor1956|Frog}} <wikitex>; {{beginproof}} {{endproof}} </wikitex> == Section == === Subsection === \label{subsection} Refert to subsection \ref{subsection} \begin{theorem} \label{thm} test \end{theorem} Refer to theorem \ref{thm} == Section == <wikitex>; An [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]]. Another <ref> Test1 </ref> [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]]. </wikitex> dfa<ref>Test2</ref> ==Footnotes== <references/><!--<bibitemswithcorrections/>--> <!-- {{#dpl: |namespace=Template |titlematch=%Milnor% |resultsheader=<h3 id="letter:{{{1}}}">{{{1}}}</h3> |noresultsheader=<div style="display:none"></div> |shownamespace=false |mode=userformat |listseparators=,\n* [[%PAGE%|[%TITLE%]]] ,, |includepage=* |ordermethod=title }} --> == Images == <wikitex>; {{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}} {{beginthm|Theorem}} For a good exposition of Theorem \ref{thm:2.1} see also \cite{Rourke&Sanderson1972a|p. 63}. {{endthm}} {{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]] \begin{theorem} \label{thm:1} We have $f \colon X \to Y$ \end{theorem} Reference \ref{thm:1} By Theorem {{equation|$\alpha$|1.2}} {{eqref|1.2}} {{begineq|eqtest|$\alpha$}}<label>qtest</label>{{endeq}} {{beginthm|Theorem}} Frog {{endthm}} \ref{qtest} \ref{eqtest} Here is some text leading up to an equation {{beginrem| }} $$ A = B $$ {{endrem}} Here is some more text after the equation to see how it looks. Here is some text leading up to an equation $$ A = B $$ Here is some more text after the equation to see how it looks. :{| border="1" cellpadding="2" class="wikitable" style="text-align:center" |- ! 4k !! 8 !! 12 !! 16 !! 20 !! 24 !! 28 !! 32 |- ! order bP_4k | 2².7 || 2⁵.31 || 2⁶.127|| 2⁹.511|| 2¹⁰.2047.691 || 2¹³.8191 || 2¹⁴.16384.3617 |- |} <!-- !! 36 !! 40 !! 44 !! 48 !! 52 !! 56 !! 60 !! 64 |- || 2. || 2. || 2. || 2. || 2. || 2. || 2. || 2. || --> :{| border="1" cellpadding="2" class="wikitable" style="text-align:center" |- ! k !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 |- ! B_k | 1/6 || 1/30 || 1/42 || 1/30 || 5/66 || 691/2730 || 7/6 || 3617/510 || |- |} :{| border="1" cellpadding="2" class="wikitable" style="text-align:center" |- ! Dim n !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20 |- ! order Θ_''n'' | 1 || 1 || 1 || 1 || 1 || 1 || 28 || 2 || 8 || 6 || 992 || 1 || 3 || 2 || 16256 || 2 || 16 || 16 || 523264 || 24 |- !''bP''_''n''+1 | 1 || 1 || 1 || 1 || 1 || 1 || 28 || 1 || 2 || 1 || 992 || 1 || 1 || 1 || 8128 || 1 || 2 || 1 || 261632 || 1 |- !Θ_''n''/''bP''_''n''+1 | 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||2×2 || 6 || 1 || 1 || 3 || 2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24 |- !π_''n''^''S''/''J'' | 1 || 2 || 1 || 1 || 1 || 2 || 1 || 2 || 2×2 || 6 || 1 || 1 || 3 || 2×2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24 |- !index | - || 2 || - || - || - || 2 || - || - || - || - || - || - || - || 2 || - || - || - || - || - || - |} [[Media:Spaceform.tiff|link text]] $$ f \colon X \to Y $$ <DPL> namespace=Main nottitlematch=Main_Page|Sandbox|Links ordermethod=lastedit order=descending minoredits=exclude author=%Diarmuid Crowley% count=10 </DPL> Just a fest $f \colon A \to B$. $\Q$ {{beginthm|a theorem}} \label{thma} {{endthm}} $\text{Spin}$ by theorem \ref{thma} <ol start="9"> <li>Amsterdam</li> <li>Rotterdam</li> <li>The Hague</li> </ol> <nowiki>[</nowiki>[[#{{anchorencode:Mess1990}}|Mess1990]]<nowiki>]</nowiki>{{#RefAdd:Mess1990}} <!-- $\sf{no serifs please}$ --> $\left( \begin{array}{ll} \alpha & \beta \ \gamma & \delta \end{array} \right)$ $f = T$ $ f : X \to Y$ $$ f : X \to Y $$ $\Ker$ $\mathscr{A}$ $\mathscr{B}$ \bf{ bold} \it{italic} \em{emphasis} </wikitex> [[Image:Foliation.png|thumb|300px|3-dimensional Reeb foliation]] == Tests == {{citeD|Ranicki1981}} {{cite|Milnor1956}} {{cite|Milnor1956|Theorem 1}} {{citeD|Milnor1956}} {{citeD|Milnor1956|Theorem 1}} {{citeD2|Milnor1956|Frog}} <wikitex>; {{beginproof}} {{endproof}} </wikitex> == Section == === Subsection === \label{subsection} Refert to subsection \ref{subsection} \begin{theorem} \label{thm} test \end{theorem} Refer to theorem \ref{thm} == Section == <wikitex>; An [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]]. Another <ref> Test1 </ref> [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]]. </wikitex> dfa<ref>Test2</ref> ==Footnotes== <references/>C^r$-differentiable $(1 \leq r \leq \infty )$$(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 3.1. 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 4.1 [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}$.

Theorem 4.2. For a good exposition of Theorem 4.1 see also [Rourke&Sanderson1972a, p. 63].

Theorem 4.3 [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 4.4. 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

Theorem 7.1. We have $f \colon X \to Y$

Reference 7.1

By Theorem

$\alpha$(1.2)

1.2

{{#addlabel: test}}

(1)$\alpha$eqtest

Theorem 7.2. Frog

1 \ref{eqtest}

Here is some text leading up to an equation

7.3. $$ A = B $$

Here is some more text after the equation to see how it looks.

Here is some text leading up to an equation $$ A = B $$ Here is some more text after the equation to see how it looks.

4k	8	12	16	20	24	28	32
order bP4k	22.7	25.31	26.127	29.511	210.2047.691	213.8191	214.16384.3617

k	1	2	3	4	5	6	7	8
Bk	1/6	1/30	1/42	1/30	5/66	691/2730	7/6	3617/510

Dim n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
order Θn	1	1	1	1	1	1	28	2	8	6	992	1	3	2	16256	2	16	16	523264	24
bPn+1	1	1	1	1	1	1	28	1	2	1	992	1	1	1	8128	1	2	1	261632	1
Θn/bPn+1	1	1	1	1	1	1	1	2	2×2	6	1	1	3	2	2	2	2×2×2	8×2	2	24
πnS/J	1	2	1	1	1	2	1	2	2×2	6	1	1	3	2×2	2	2	2×2×2	8×2	2	24
index	-	2	-	-	-	2	-	-	-	-	-	-	-	2	-	-	-	-	-	-

link text

$$ f \colon X \to Y $$

Extension DPL (warning): current configuration allows execution of DPL code from protected pages only.

Just a fest $f \colon A \to B$.

$\Q$

a theorem 7.4.

$\text{Spin}$

by theorem 7.4

9. Amsterdam
10. Rotterdam
11. The Hague

[Mess1990]

$\left( \begin{array}{ll} \alpha & \beta \\ \gamma & \delta \end{array} \right)$

$f = T$

$ f : X \to Y$

$$ f : X \to Y $$

$\Ker$

$\mathscr{A}$ $\mathscr{B}$

bold italic emphasis

</wikitex>

File:Foliation.png

3-dimensional Reeb foliation

6 Tests

[Ranicki1981] [Milnor1956] [Milnor1956, Theorem 1] [Milnor1956] [Milnor1956, Theorem 1] Frog

Proof.

$\square$$\square$

7 Section

7.1 Subsection

Refert to subsection 9.1

Theorem 9.1. test

Refer to theorem 9.1

8 Section

An inter-Wiki link.

Another ; inter-Wiki link.

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