# Sandbox

 The sandbox is the page where you can experiment with the wiki syntax. Feel free to write nonsense or clear the page whenever you want.

## 1 Images

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## Definition

We work in a fixed category CAT of topological, piecewise linear, $C^r$ == Images == ; {{Authors|Ulrich Koschorke}}{{Definition reviewed}} == Definition == ; 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 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. ==Existence of embeddings== ; {{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. ==Classification== ; 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. == 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}}{{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 bP4k | 22.7 || 25.31 || 26.127|| 29.511|| 210.2047.691 || 213.8191 || 214.16384.3617 |- |} :{| border="1" cellpadding="2" class="wikitable" style="text-align:center" |- ! 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 || |- |} :{| 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 namespace=Main nottitlematch=Main_Page|Sandbox|Links ordermethod=lastedit order=descending minoredits=exclude author=%Diarmuid Crowley% count=10 Just a fest f \colon A \to B. \Q {{beginthm|a theorem}} \label{thma} {{endthm}} \text{Spin} by theorem \ref{thma}
1. Amsterdam
2. Rotterdam
3. The Hague
[[[#{{anchorencode:Mess1990}}|Mess1990]]]{{#RefAdd: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} \bf{ bold} \it{italic} \em{emphasis}
[[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}} ; {{beginproof}} {{endproof}} == Section == === Subsection === \label{subsection} Refert to subsection \ref{subsection} \begin{theorem} \label{thm} test \end{theorem} Refer to theorem \ref{thm} == Section == ; An [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]]. Another Test1 [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]]. dfaTest2 ==Footnotes== == Images == ; {{Authors|Ulrich Koschorke}}{{Definition reviewed}} == Definition == ; 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 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. ==Existence of embeddings== ; {{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. ==Classification== ; 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. == 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}}{{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 bP4k | 22.7 || 25.31 || 26.127|| 29.511|| 210.2047.691 || 213.8191 || 214.16384.3617 |- |} :{| border="1" cellpadding="2" class="wikitable" style="text-align:center" |- ! 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 || |- |} :{| 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 namespace=Main nottitlematch=Main_Page|Sandbox|Links ordermethod=lastedit order=descending minoredits=exclude author=%Diarmuid Crowley% count=10 Just a fest f \colon A \to B. \Q {{beginthm|a theorem}} \label{thma} {{endthm}} \text{Spin} by theorem \ref{thma}
1. Amsterdam
2. Rotterdam
3. The Hague
[[[#{{anchorencode:Mess1990}}|Mess1990]]]{{#RefAdd: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} \bf{ bold} \it{italic} \em{emphasis}
[[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}} ; {{beginproof}} {{endproof}} == Section == === Subsection === \label{subsection} Refert to subsection \ref{subsection} \begin{theorem} \label{thm} test \end{theorem} Refer to theorem \ref{thm} == Section == ; An [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]]. Another Test1 [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]]. dfaTest2 ==Footnotes== 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}$.

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.