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== Images ==
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{{beginthm|Theorem|\cite{Penrose&Whitehead&Zeeman1961}}}\label{thm:2.1}
<wikitex>;
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{{Authors|Ulrich Koschorke}}{{Definition reviewed}}
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== Definition ==
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<wikitex>;
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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.
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Let $ f : M^m \rightarrow N^n $ be such a map between manifolds of the indicated dimensions $ 1 \leq m < n $.
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{{beginthm|Definition}}
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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.
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{{endthm}}
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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 -->
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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
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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 $\R^3$.
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</wikitex>
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==Existence of embeddings==
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<wikitex>;
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{{beginthm|Theorem|aaaa}}\label{thm:2.1}
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For every compact $m$--dimensional PL-manifold $M$ there exists a PL--embedding $ M \hookrightarrow \R^{2m}$.
For every compact $m$--dimensional PL-manifold $M$ there exists a PL--embedding $ M \hookrightarrow \R^{2m}$.
{{endthm}}
{{endthm}}
{{beginthm|Theorem|sadfsda}}
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{{beginrem|Remark}}
For a good exposition of Theorem \ref{thm:2.1} see also \cite{Rourke&Sanderson1972a|p. 63}.
For a good exposition of Theorem \ref{thm:2.1} see also \cite{Rourke&Sanderson1972a|p. 63}.
{{endthm}}
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{{endrem}}
{{beginthm|Theorem|daf}}\label{thm:2.2}
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{{beginthm|Theorem|\cite{Whitney1944}}}\label{thm:2.2}
\cite{Whitney1944}
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For every closed m--dimensional $C^{\infty}$--manifold $M$ there exists a $C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$.
For every closed m--dimensional $C^{\infty}$--manifold $M$ there exists a $C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$.
{{endthm}}
{{endthm}}
{{beginrem|Remark|asdfasd}}
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{{beginrem|Remark}}
For a more modern exposition see also \cite{Adachi1993|p. 67ff}.
For a more modern exposition see also \cite{Adachi1993|p. 67ff}.
{{endrem}}
{{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}
\begin{theorem} \label{thm:1}

Revision as of 07:11, 25 April 2013

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.


Theorem 0.1 [Penrose&Whitehead&Zeeman1961]. For every compact $m$--dimensional PL-manifold $M$ there exists a PL--embedding $ M \hookrightarrow \R^{2m}$.

Remark 0.2. For a good exposition of Theorem 0.1 see also [Rourke&Sanderson1972a, p. 63].

Theorem 0.3 [Whitney1944]. For every closed m--dimensional $C^{\infty}$--manifold $M$ there exists a $C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$.

Remark 0.4. For a more modern exposition see also [Adachi1993, p. 67ff].

Theorem 0.5. We have $f \colon X \to Y$

Reference 0.5

By Theorem

$\alpha$(1.2)

1.2

{{#addlabel: test}}

(1)$\alpha$eqtest


Theorem 0.6. Frog

1 \ref{eqtest}

Here is some text leading up to an equation

0.7. $$ A = B $$

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 0.8.

$\text{Spin}$

by theorem 0.8

  1. Amsterdam
  2. Rotterdam
  3. 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

Contents

1 Tests

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

Proof.

\square

2 Section

2.1 Subsection

Refert to subsection 2.1

Theorem 2.1. test

Refer to theorem 2.1

3 Section

An inter-Wiki link.

Another [1]; inter-Wiki link.


dfa[2]

4 Footnotes

  1. Test1
  2. Test2
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