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<wikitex refresh">;
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<!--<bibitemswithcorrections/>-->
* For the more general case where $A H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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<!-- {{#dpl:
</wikitex>
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|namespace=Template
<wikitex refresh include="MediaWiki:MathFontCM">;
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|titlematch=%Milnor%
* For the more general case where $B H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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|resultsheader=<h3 id="letter:{{{1}}}">{{{1}}}</h3>
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|noresultsheader=<div style="display:none"></div>
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|shownamespace=false
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|listseparators=,\n* [[%PAGE%|[%TITLE%]]] ,,
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}} -->
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<stepsectioncounter/>
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== File Creation Practice ==
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[[Media:New_submission.pdf|Click here to access the pdf file]].
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== Testing equation numbering ==
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<wikitex>;
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\begin{equation} \label{eq:1} A = B \end{equation}
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Here is a reference to equation \ref{eq:1}
</wikitex>
</wikitex>
<wikitex refresh include="MediaWiki:MathFontHV">;
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== Still testing equation numbering ==
* For the more general case where $C H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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<wikitex>;
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\begin{equation}\label{test} C = D \end{equation}
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\ref{test}
</wikitex>
</wikitex>
<wikitex refresh include="MediaWiki:MathFont">;
* For the more general case where $E H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
</wikitex>
<wikitex refresh include="MediaWiki:MathFontLM">;
* For the more general case where $F H_1(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
</wikitex>
== Introduction ==
<wikitex refresh include="MediaWiki:MathFontHV">;
Let $\mathcal{M}_6(0)$ be the set of diffeomorphism classes of [[wikipedia:Closed_manifold|closed]] [[wikipedia:Differentiable_manifold|smooth]] [[wikipedia:Simply-connected|simply-connected]] 2-connected 6-manifolds $M$.
The classification $\mathcal{M}_6(0)$ was one of Smale's first applications of the [[Wikipedia:h-cobordism_theorem|h-cobordism]] theorem {{cite|Smale1962a|Corollary 1.3}}. The classification, as for [[Surface|oriented surfaces]] is strikingly simple: every 2-connected 6-manifold $M$ is diffeomorphic to a [[Wikipedia:Connected-sum|connected-sum]]
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== Lists==
$$ M \cong \sharp_r(S^3 \times S^3)$$
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<wikitex>;
where by definition $\sharp_0(S^3 \times S^3) = S^6$ and in general $r$ is determined by the formula for the [[Wikipedia:Euler characteristic|Euler characteristic]] of $M$
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<ol style="list-tsyle-type:lower-roman>
$$ \chi(M) = 2 - 2r.$$
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<li>Frog</li>
* For the more general case where $H_2(M) \neq 0$, see [[6-manifolds: 1-connected|6-manifolds: 1-connected]].
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</ol>
== Construction and examples ==
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$$ A \xrightarrow{f} B$$
The following gives a complete list of 2-connected 6-manifolds up to diffeomorphism:
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* $S^6$, the standard 6-sphere.
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* $\sharp_b(S^3 \times S^3)$, the $b$-fold connected sum of $S^3 \times S^3$.
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== Invariants ==
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$--$
Suppose that $M$ is diffeomorphic to $\sharp_b(S^3 \times S^3)$ then:
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</wikitex>
* $\pi_3(M) \cong H_3(M) \cong \Zz^{2b}$,
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{{beginthm|Theorem|\cite{Penrose&Whitehead&Zeeman1961}}}\label{thm:2.1}
* the third Betti-number of $M$ is given by $b_3(M) = 2b$,
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For every compact $m$--dimensional PL-manifold $M$ there exists a PL--embedding $ M \hookrightarrow \R^{2m}$.
* the Euler characteristic of $M$ is given by $\chi(M) = 2 - 2b$,
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{{endthm}}
* the [[Intersection forms|intersection form]] of $M$ is isomorphic to the sum of b-copies of $H_{-}(\Zz)$, the standard skew-symmetric hyperbolic form on $\Zz^2$.
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{{beginrem|Remark}}
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For a good exposition of Theorem \ref{thm:2.1} see also \cite{Rourke&Sanderson1972a|p. 63}.
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{{endrem}}
== Classification ==
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{{beginthm|Theorem|\cite{Whitney1944}}}\label{thm:2.2}
Recall that the following theorem was stated in other words in the introduction:
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For every closed m--dimensional $C^{\infty}$--manifold $M$ there exists a $C^{\infty}$--embedding $M \hookrightarrow \R^{2m}$.
{{beginthm|Theorem|{{cite|Smale1962a|Corolary 1.3}}}} \label{thm:classification}
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The semi-group of 2-connected 6-manifolds is generated by $S^3 \times S^3$.
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{{endthm}}
{{endthm}}
Hence if $\Nn$ denotes the natural numbers we obtain a bijection
$$ \mathcal{M}_6(0)\equiv \Nn,~~~[M] \mapsto \frac{1}{2}b_3(M).$$
== Further discussion ==
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{{beginrem|Remark}}
=== Topological 2-connected 6-manifolds ===
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For a more modern exposition see also \cite{Adachi1993|p. 67ff}.
Let $\mathcal{M}^{\Top}_6(e)$ be the set of homeomorphism classes of topological 2-connected 6-manifolds.
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{{endrem}}
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\begin{theorem} \label{thm:1}
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We have $f \colon X \to Y$
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\end{theorem}
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Reference \ref{thm:1}
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By Theorem
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{{equation|$\alpha$|1.2}}
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{{eqref|1.2}}
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{{begineq|eqtest|$\alpha$}}<label>qtest</label>{{endeq}}
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{{beginthm|Theorem}}
{{beginthm|Theorem}}
Every topological 2-connected 6-manifold admits a smooth structure which is unique up to diffoemorphism. That is, there is a bijection
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Frog
$$ \mathcal{M}_6(e) \rightarrow \mathcal{M}^{\Top}_6(e).$$
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{{endthm}}
{{endthm}}
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\ref{qtest}
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\ref{eqtest}
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Here is some text leading up to an equation
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{{beginrem| }}
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$$ A = B $$
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{{endrem}}
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Here is some more text after the equation to see how it looks.
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Here is some text leading up to an equation
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$$ A = B $$
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Here is some more text after the equation to see how it looks.
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:{| border="1" cellpadding="2" class="wikitable" style="text-align:center"
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|-
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! 4k !! 8 !! 12 !! 16 !! 20 !! 24 !! 28 !! 32
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|-
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! order bP<sub>4k</sub>
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| 2<sup>2</sup>.7 || 2<sup>5</sup>.31 || 2<sup>6</sup>.127|| 2<sup>9</sup>.511|| 2<sup>10</sup>.2047.691 || 2<sup>13</sup>.8191 || 2<sup>14</sup>.16384.3617
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|-
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|}
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<!-- !! 36 !! 40 !! 44 !! 48 !! 52 !! 56 !! 60 !! 64 |-
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|| 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. || 2<sup></sup>. ||
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-->
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:{| border="1" cellpadding="2" class="wikitable" style="text-align:center"
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|-
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! k !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8
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|-
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! B<sub>k</sub>
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| 1/6 || 1/30 || 1/42 || 1/30 || 5/66 || 691/2730 || 7/6 || 3617/510 ||
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|-
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|}
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:{| border="1" cellpadding="2" class="wikitable" style="text-align:center"
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|-
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! Dim n !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20
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|-
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! order Θ<sub>''n''</sub>
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| 1 || 1 || 1 || 1 || 1 || 1 || 28 || 2 || 8 || 6 || 992 || 1 || 3 || 2 || 16256 || 2 || 16 || 16 || 523264 || 24
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|-
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!''bP''<sub>''n''+1</sub>
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| 1 || 1 || 1 || 1 || 1 || 1 || 28 || 1 || 2 || 1 || 992 || 1 || 1 || 1 || 8128 || 1 || 2 || 1 || 261632 || 1
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|-
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!Θ<sub>''n''</sub>/''bP''<sub>''n''+1</sub>
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| 1 || 1 || 1 || 1 || 1 || 1 || 1 || 2 ||2×2 || 6 || 1 || 1 || 3 || 2 || 2 || 2 || 2×2×2 || 8×2 || 2 || 24
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|-
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!π<sub>''n''</sub><sup>''S''</sup>/''J''
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| 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
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|-
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!index
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| - || 2 || - || - || - || 2 || - || - || - || - || - || - || - || 2 || - || - || - || - || - || -
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|}
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[[Media:Spaceform.tiff|link text]]
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$$ f \colon X \to Y $$
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<DPL>
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namespace=Main
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nottitlematch=Main_Page|Sandbox|Links
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ordermethod=lastedit
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order=descending
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minoredits=exclude
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author=%Diarmuid Crowley%
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count=10
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</DPL>
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Just a fest $f \colon A \to B$.
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$\Q$
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{{beginthm|a theorem}}
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\label{thma}
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{{endthm}}
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$\text{Spin}$
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by theorem \ref{thma}
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<ol start="9">
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<li>Amsterdam</li>
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<li>Rotterdam</li>
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<li>The Hague</li>
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</ol>
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<nowiki>[</nowiki>[[#{{anchorencode:Mess1990}}|Mess1990]]<nowiki>]</nowiki>{{#RefAdd:Mess1990}}
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<!-- $\sf{no serifs please}$ -->
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$\left( \begin{array}{ll} \alpha & \beta \\ \gamma & \delta \end{array} \right)$
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$f = T$
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$ f : X \to Y$
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$$ f : X \to Y $$
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$\Ker$
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$\mathscr{A}$ $\mathscr{B}$
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\bf{ bold} \it{italic} \em{emphasis}
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</wikitex>
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[[Image:Foliation.png|thumb|300px|3-dimensional Reeb foliation]]
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== Tests ==
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{{citeD|Ranicki1981}}
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{{cite|Milnor1956}} {{cite|Milnor1956|Theorem 1}} {{citeD|Milnor1956}} {{citeD|Milnor1956|Theorem 1}} {{citeD2|Milnor1956|Frog}}
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<wikitex>;
{{beginproof}}
{{beginproof}}
For any such manifold $M$ we have $H^4(M; \Zz/2) \cong 0$ and so $M$ is smoothable (see [[6-manifolds: 1-connected#Smoothing theory|6-manifolds: 1-connected]]). Any two homeomorphic manifolds have the same Euler Characteristic and so by Theorem \ref{thm:classification} are diffeomorphic.
{{endproof}}
{{endproof}}
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</wikitex>
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== Section ==
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=== Subsection ===
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\label{subsection}
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Refert to subsection \ref{subsection}
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\begin{theorem} \label{thm}
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test
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\end{theorem}
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Refer to theorem \ref{thm}
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== Section ==
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<wikitex>;
+
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'''$\textup{CW}_0$'''
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An [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]].
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Another <ref> Test1 </ref> [[map-eds:Manifold Atlas:Projects in the Atlas|inter-Wiki link]].
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dfa<ref>Test2</ref>
=== Mapping class groups ===
...
</wikitex>
</wikitex>
== References ==
+
==Footnotes==
+
<references/>
+
==References==
{{#RefList:}}
{{#RefList:}}
[[Category:Manifolds]]
[[Category:Highly-connected manifolds]]

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

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