Sandbox

From Manifold Atlas
(Difference between revisions)
Jump to: navigation, search
m
m
(26 intermediate revisions by 2 users not shown)
Line 12: Line 12:
}} -->
}} -->
== Images ==
+
<stepsectioncounter/>
<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 $.
+
== File Creation Practice ==
{{beginthm|Definition}}
+
[[Media:New_submission.pdf|Click here to access the pdf file]].
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 -->
+
== Testing equation numbering ==
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
+
<wikitex>;
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 $\R^3$.
+
\begin{equation} \label{eq:1} A = B \end{equation}
+
Here is a reference to equation \ref{eq:1}
+
</wikitex>
+
== Still testing equation numbering ==
+
<wikitex>;
+
\begin{equation}\label{test} C = D \end{equation}
+
\ref{test}
</wikitex>
</wikitex>
==Existence of embeddings==
+
== Lists==
<wikitex>;
<wikitex>;
+
<ol style="list-tsyle-type:lower-roman>
+
<li>Frog</li>
+
</ol>
{{beginthm|Theorem|aaaa}}\label{thm:2.1}
+
$$ A \xrightarrow{f} B$$
+
+
$--$
+
</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}$.
For every compact $m$--dimensional PL-manifold $M$ there exists a PL--embedding $ M \hookrightarrow \R^{2m}$.
{{endthm}}
{{endthm}}
{{beginthm|Theorem|sadfsda}}
+
{{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}}
+
{{endrem}}
{{beginthm|Theorem|\cite{Whitney1944}}}\label{thm:2.2}
{{beginthm|Theorem|\cite{Whitney1944}}}\label{thm:2.2}
Line 45: Line 51:
{{endthm}}
{{endthm}}
{{beginrem|Remark|asdfasd}}
+
{{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}
Line 216: Line 207:
== Section ==
== Section ==
<wikitex>;
<wikitex>;
+
+
'''$\textup{CW}_0$'''
An [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]].
An [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]].
Line 221: Line 214:
Another <ref> Test1 </ref> [[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>
dfa<ref>Test2</ref>
+
+
</wikitex>
==Footnotes==
==Footnotes==
<references/>
<references/>
+
==References==
+
{{#RefList:}}

Latest revision as of 07:31, 15 October 2019

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

[edit] 1 File Creation Practice

Click here to access the pdf file.

[edit] 2 Testing equation numbering

(1)A = B

Here is a reference to equation 1

[edit] 3 Still testing equation numbering

(2)C = D

2

[edit] 4 Lists

  1. Frog
\displaystyle  A \xrightarrow{f} B

--

Theorem 5.1 [Penrose&Whitehead&Zeeman1961]. For every compact $m$--dimensional PL-manifold $M$ there exists a PL--embedding $ M \hookrightarrow \R^{2m}$.

Remark 5.2. For a good exposition of Theorem 5.1 see also [Rourke&Sanderson1972a, p. 63].

Theorem 5.3 [Whitney1944]. For every closed m--dimensional $C^{\infty}$--manifold $M$ there exists a $C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$.

Remark 5.4. For a more modern exposition see also [Adachi1993, p. 67ff].

Theorem 5.5. We have $f \colon X \to Y$

Reference 5.5

By Theorem

$\alpha$(1.2)

1.2

{{#addlabel: test}}

(3)$\alpha$eqtest


Theorem 5.6. Frog

3 \ref{eqtest}

Here is some text leading up to an equation

5.7. $$ A = B $$

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 5.8.

$\text{Spin}$

by theorem 5.8

  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

[edit] 5 Tests

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

Proof.

\square

[edit] 6 Section

[edit] 6.1 Subsection

Refert to subsection 7.1

Theorem 7.1. test

Refer to theorem 7.1

[edit] 7 Section

\textup{CW}_0

An inter-Wiki link.

Another [1]; inter-Wiki link.

dfa[2]


[edit] 8 Footnotes

  1. Test1
  2. Test2

[edit] 9 References

Personal tools
Namespaces
Variants
Actions
Navigation
Interaction
Toolbox