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<p><a id="X855CD0808058727D" name="X855CD0808058727D"></a></p>
<div class="ChapSects"><a href="chap30.html#X855CD0808058727D">30 <span class="Heading">Regular CW-Complexes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap30.html#X7CFDEEC07F15CF82">30.1 <span class="Heading">  </span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X825EF11A86624E44">30.1-1 SimplicialComplexToRegularCWComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X8723DF8A7AA8AE9E">30.1-2 CubicalComplexToRegularCWComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X802E1927807969C1">30.1-3 CriticalCellsOfRegularCWComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7A1C427578108B7E">30.1-4 ChainComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X8307D5A57EFAC8EE">30.1-5 ChainComplexOfRegularCWComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap30.html#X7EAE7E4181546C17">30.1-6 FundamentalGroup</a></span>
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<h3>30 <span class="Heading">Regular CW-Complexes</span></h3>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>30.1 <span class="Heading">  </span></h4>

<p><a id="X825EF11A86624E44" name="X825EF11A86624E44"></a></p>

<h5>30.1-1 SimplicialComplexToRegularCWComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SimplicialComplexToRegularCWComplex</code>( <var class="Arg">K</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a simplicial complex <span class="SimpleMath">K</span> and returns the corresponding regular CW-complex.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRandomComplexes.html">2</a></span> </p>

<p><a id="X8723DF8A7AA8AE9E" name="X8723DF8A7AA8AE9E"></a></p>

<h5>30.1-2 CubicalComplexToRegularCWComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CubicalComplexToRegularCWComplex</code>( <var class="Arg">K</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CubicalComplexToRegularCWComplex</code>( <var class="Arg">K</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a pure cubical complex (or cubical complex) <span class="SimpleMath">K</span> and returns the corresponding regular CW-complex. If a positive integer <span class="SimpleMath">n</span> is entered as an optional second argument, then just the <span class="SimpleMath">n</span>-skeleton of <span class="SimpleMath">K</span> is returned.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">1</a></span> </p>

<p><a id="X802E1927807969C1" name="X802E1927807969C1"></a></p>

<h5>30.1-3 CriticalCellsOfRegularCWComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CriticalCellsOfRegularCWComplex</code>( <var class="Arg">Y</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CriticalCellsOfRegularCWComplex</code>( <var class="Arg">Y</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a regular CW-complex <span class="SimpleMath">Y</span> and returns the critical cells of <span class="SimpleMath">Y</span> with respect to some discrete vector field. If <span class="SimpleMath">Y</span> does not initially have a discrete vector field then one is constructed.</p>

<p>If a positive integer <span class="SimpleMath">n</span> is given as a second optional input, then just the critical cells in dimensions up to and including <span class="SimpleMath">n</span> are returned.</p>

<p>The function <span class="SimpleMath">CriticalCellsOfRegularCWComplex(Y)</span> works by homotopy reducing cells starting at the top dimension. The function <span class="SimpleMath">CriticalCellsOfRegularCWComplex(Y,n)</span> works by homotopy coreducing cells starting at dimension 0. The two methods may well return different numbers of cells.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutPeripheral.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRandomComplexes.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">3</a></span> </p>

<p><a id="X7A1C427578108B7E" name="X7A1C427578108B7E"></a></p>

<h5>30.1-4 ChainComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChainComplex</code>( <var class="Arg">Y</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a regular CW-complex <span class="SimpleMath">Y</span> and returns the cellular chain complex of a CW-complex W whose cells correspond to the critical cells of <span class="SimpleMath">Y</span> with respect to some discrete vector field. If <span class="SimpleMath">Y</span> does not initially have a discrete vector field then one is constructed.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap3.html">2</a></span> , <span class="URL"><a href="../tutorial/chap4.html">3</a></span> , <span class="URL"><a href="../tutorial/chap10.html">4</a></span> , <span class="URL"><a href="../tutorial/chap12.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutMetrics.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutBredon.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCubical.html">11</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">12</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">13</a></span> </p>

<p><a id="X8307D5A57EFAC8EE" name="X8307D5A57EFAC8EE"></a></p>

<h5>30.1-5 ChainComplexOfRegularCWComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChainComplexOfRegularCWComplex</code>( <var class="Arg">Y</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a regular CW-complex <span class="SimpleMath">Y</span> and returns the cellular chain complex of <span class="SimpleMath">Y</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> </p>

<p><a id="X7EAE7E4181546C17" name="X7EAE7E4181546C17"></a></p>

<h5>30.1-6 FundamentalGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FundamentalGroup</code>( <var class="Arg">Y</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FundamentalGroup</code>( <var class="Arg">Y</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a regular CW-complex <span class="SimpleMath">Y</span> and, optionally, the number of some 0-cell. It returns the fundamental group of <span class="SimpleMath">Y</span> based at the 0-cell <span class="SimpleMath">n</span>. The group is returned as a finitely presented group. If <span class="SimpleMath">n</span> is not specified then it is set <span class="SimpleMath">n=1</span>. The algorithm requires a discrete vector field on <span class="SimpleMath">Y</span>. If <span class="SimpleMath">Y</span> does not initially have a discrete vector field then one is constructed.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap2.html">2</a></span> , <span class="URL"><a href="../tutorial/chap3.html">3</a></span> , <span class="URL"><a href="../tutorial/chap4.html">4</a></span> , <span class="URL"><a href="../tutorial/chap5.html">5</a></span> , <span class="URL"><a href="../tutorial/chap11.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLinks.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPeripheral.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoveringSpaces.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoverinSpaces.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutQuandles.html">11</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRandomComplexes.html">12</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutKnots.html">13</a></span> </p>


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