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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}.
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For a more modern exposition see also \cite{Adachi1993|p. 67ff}.
For a more modern exposition see also \cite{Adachi1993|p. 67ff}.
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Revision as of 16:30, 24 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.


Contents

1 Images

The user responsible for this page is Ulrich Koschorke. No other user may edit this page at present.

This page is being independently reviewed under the supervision of the Editorial Board.

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.

Definition 3.1. We call f an embedding (and we write f : M \hookrightarrow N) if f is an immersion which maps 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 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.

2 Existence of embeddings

Theorem 4.1 [Penrose&Whitehead&Zeeman1961]. For every compact m--dimensional PL-manifold M there exists a PL--embedding 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}--manifold M there exists a C^{\infty}--embedding 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 are valid also in the categories of real analytic maps and of isometrics (Nash) when N \gg 2m is sufficiently high.

3 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 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.

4 References

5 External links


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

  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

6 Tests

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

Proof.

\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 [1]; inter-Wiki link.


dfa[2]

9 Footnotes

  1. Test1
  2. Test2
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