# Sandbox

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− | + | == File Creation Practice == | |

− | + | [[Media:New_submission.pdf|Click here to access the pdf file]]. | |

== Testing equation numbering == | == Testing equation numbering == |

## Latest revision as of 08:31, 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

Here is a reference to equation 1

## [edit] 3 Still testing equation numbering

## [edit] 4 Lists

- Frog

**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

**(**1.2

**)**

{{#addlabel: test}}

**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 bP _{4k}2 ^{2}.72 ^{5}.312 ^{6}.1272 ^{9}.5112 ^{10}.2047.6912 ^{13}.81912 ^{14}.16384.3617

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

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 - - - - - -

$$ f \colon X \to Y $$

*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

- Amsterdam
- Rotterdam
- 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>

## [edit] 5 Tests

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

**Proof.**

## [edit] 6 Section

### [edit] 6.1 Subsection

Refert to subsection 7.1

**Theorem 7.1.**
test

Refer to theorem 7.1

## [edit] 7 Section

**
**

An inter-Wiki link.

Another ^{[1]}; inter-Wiki link.

dfa^{[2]}

## [edit] 8 Footnotes

## [edit] 9 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 - [Mess1990] G. Mess,
*Examples of Poincaré duality groups*, Proc. Amer. Math. Soc.**110**(1990), no.4, 1145–1146. MR1019274 (91c:20075) Zbl 0709.57025 - [Milnor1956] J. Milnor,
*On manifolds homeomorphic to the -sphere*, Ann. of Math. (2)**64**(1956), 399–405. MR0082103 (18,498d) Zbl 0072.18402 - [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 -manifold in -space*, Ann. of Math. (2)**45**(1944), 220–246. MR0010274 (5,273g) Zbl 0063.08237