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<p><a id="X8213E6467969C33F" name="X8213E6467969C33F"></a></p>
<div class="ChapSects"><a href="chap27.html#X8213E6467969C33F">27 <span class="Heading">Torsion Subcomplexes</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap27.html#X7CFDEEC07F15CF82">27.1 <span class="Heading">  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X802BB08884521CA0">27.1-1 RigidFacetsSubdivision</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X8643001B8648B111">27.1-2 IsPNormal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7EAD6CFB84E5E01E">27.1-3 TorsionSubcomplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X86367F26851D7297">27.1-4 DisplayAvailableCellComplexes</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X781ABD9A86F77B3B">27.1-5 VisualizeTorsionSkeleton</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X87D2C8A1873EDA4C">27.1-6 ReduceTorsionSubcomplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7FAA66787B39D4FF">27.1-7 EquivariantEulerCharacteristic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7C0EF3F17C513473">27.1-8 CountingCellsOfACellComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X803D6BE97E9FD89A">27.1-9 CountingControlledSubdividedCells</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X786031BF85CFA3EA">27.1-10 CountingBaryCentricSubdividedCells</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X7F6DE2CA7F8BCA5E">27.1-11 EquivariantSpectralSequencePage</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap27.html#X832CEB637DC26E33">27.1-12 ExportHapCellcomplexToDisk</a></span>
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<h3>27 <span class="Heading">Torsion Subcomplexes</span></h3>

<p>The Torsion Subcomplex subpackage has been conceived and implemented by <strong class="button">Bui Anh Tuan</strong> and <strong class="button"> Alexander D. Rahm</strong>.</p>

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

<h4>27.1 <span class="Heading">  </span></h4>

<p><a id="X802BB08884521CA0" name="X802BB08884521CA0"></a></p>

<h5>27.1-1 RigidFacetsSubdivision</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; RigidFacetsSubdivision</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>It inputs an <span class="SimpleMath">n</span>-dimensional <span class="SimpleMath">G</span>-equivariant CW-complex <span class="SimpleMath">X</span> on which all the cell stabilizer subgroups in <span class="SimpleMath">G</span> are finite. It returns an <span class="SimpleMath">n</span>-dimensional <span class="SimpleMath">G</span>-equivariant CW-complex <span class="SimpleMath">Y</span> which is topologically the same as <span class="SimpleMath">X</span>, but equipped with a <span class="SimpleMath">G</span>-CW-structure which is rigid.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X8643001B8648B111" name="X8643001B8648B111"></a></p>

<h5>27.1-2 IsPNormal</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPNormal</code>( <var class="Arg">G</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">G</span> and a prime <span class="SimpleMath">p</span>. Checks if the group G is p-normal for the prime p. Zassenhaus defines a finite group to be p-normal if the center of one of its Sylow p-groups is the center of every Sylow p-group in which it is contained.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7EAD6CFB84E5E01E" name="X7EAD6CFB84E5E01E"></a></p>

<h5>27.1-3 TorsionSubcomplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TorsionSubcomplex</code>( <var class="Arg">C</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a cell complex with action of a group as a variable or a group name. In HAP, presently the following cell complexes with stabilisers fixing their cells pointwise are available, specified by the following "groupName" strings: <br /> <br /> "SL(2,O-2)" , "SL(2,O-7)" , "SL(2,O-11)" , "SL(2,O-19)" , "SL(2,O-43)" , "SL(2,O-67)" , "SL(2,O-163)", <br /> <br /> where the symbol O[-m] stands for the ring of integers in the imaginary quadratic number field Q(sqrt(-m)), the latter being the extension of the field of rational numbers by the square root of minus the square-free positive integer m. The additive structure of this ring O[-m] is given as the module Z[omega] over the natural integers Z with basis {1, omega}, and omega being the square root of minus m if m is congruent to 1 or 2 modulo four; else, in the case m congruent 3 modulo 4, the element omega is the arithmetic mean with 1, namely <span class="SimpleMath">(1+sqrt(-m))/2</span>. <br /> <br /> The function TorsionSubcomplex prints the cells with p-torsion in their stabilizer on the screen and returns the incidence matrix of the 1-skeleton of this cellular subcomplex, as well as a Boolean value on whether the cell complex has its cell stabilisers fixing their cells pointwise. <br /> <br /> It is also possible to input the cell complexes <br /> <br /> "SL(2,Z)" , "SL(3,Z)" , "PGL(3,Z[i])" , "PGL(3,Eisenstein_Integers)" , "PSL(4,Z)" , "PSL(4,Z)_b" , "PSL(4,Z)_c" , "PSL(4,Z)_d" , "Sp(4,Z)" <br /> <br /> provided by <strong class="button">Mathieu Dutour</strong>.</p>

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

<p><a id="X86367F26851D7297" name="X86367F26851D7297"></a></p>

<h5>27.1-4 DisplayAvailableCellComplexes</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DisplayAvailableCellComplexes</code>(  )</td><td class="tdright">( function )</td></tr></table></div>
<p>Displays the cell complexes that are available in HAP.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X781ABD9A86F77B3B" name="X781ABD9A86F77B3B"></a></p>

<h5>27.1-5 VisualizeTorsionSkeleton</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; VisualizeTorsionSkeleton</code>( <var class="Arg">groupName</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Executes the function TorsionSubcomplex( groupName, p) and visualizes its output, namely the incidence matrix of the 1-skeleton of the p-torsion subcomplex, as a graph.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X87D2C8A1873EDA4C" name="X87D2C8A1873EDA4C"></a></p>

<h5>27.1-6 ReduceTorsionSubcomplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ReduceTorsionSubcomplex</code>( <var class="Arg">C</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function start with the same operations as the function TorsionSubcomplex( C, p), and if the cell stabilisers are fixing their cells pointwise, it continues as follows. <br /> <br /> It prints on the screen which cells to merge and which edges to cut off in order to reduce the p-torsion subcomplex without changing the equivariant Farrell cohomology. Finally, it prints the representative cells, their stabilizers and the Abelianization of the latter.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7FAA66787B39D4FF" name="X7FAA66787B39D4FF"></a></p>

<h5>27.1-7 EquivariantEulerCharacteristic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EquivariantEulerCharacteristic</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>It inputs an <span class="SimpleMath">n</span>-dimensional <span class="SimpleMath">Γ</span>-equivariant CW-complex <span class="SimpleMath">X</span> all the cell stabilizer subgroups in <span class="SimpleMath">Γ</span> are finite. It returns the equivariant euler characteristic obtained by using mass formula <span class="SimpleMath">∑_σ(-1)^dimσfrac1card(Γ_σ)</span></p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7C0EF3F17C513473" name="X7C0EF3F17C513473"></a></p>

<h5>27.1-8 CountingCellsOfACellComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CountingCellsOfACellComplex</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>It inputs an <span class="SimpleMath">n</span>-dimensional <span class="SimpleMath">Γ</span>-equivariant CW-complex <span class="SimpleMath">X</span> on which all the cell stabilizer subgroups in <span class="SimpleMath">Γ</span> are finite. It returns the number of cells in <span class="SimpleMath">X</span></p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X803D6BE97E9FD89A" name="X803D6BE97E9FD89A"></a></p>

<h5>27.1-9 CountingControlledSubdividedCells</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CountingControlledSubdividedCells</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>It inputs an <span class="SimpleMath">n</span>-dimensional <span class="SimpleMath">Γ</span>-equivariant CW-complex <span class="SimpleMath">X</span> on which all the cell stabilizer subgroups in <span class="SimpleMath">Γ</span> are finite. It returns the number of cells in <span class="SimpleMath">X</span> appear during the subdivision process using the RigidFacetsSubdivision.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X786031BF85CFA3EA" name="X786031BF85CFA3EA"></a></p>

<h5>27.1-10 CountingBaryCentricSubdividedCells</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CountingBaryCentricSubdividedCells</code>( <var class="Arg">X</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>It inputs an <span class="SimpleMath">n</span>-dimensional <span class="SimpleMath">Γ</span>-equivariant CW-complex <span class="SimpleMath">X</span> on which all the cell stabilizer subgroups in <span class="SimpleMath">Γ</span> are finite. It returns the number of cells in <span class="SimpleMath">X</span> appear during the subdivision process using the barycentric subdivision.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X7F6DE2CA7F8BCA5E" name="X7F6DE2CA7F8BCA5E"></a></p>

<h5>27.1-11 EquivariantSpectralSequencePage</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EquivariantSpectralSequencePage</code>( <var class="Arg">C</var>, <var class="Arg">m</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>It inputs a triple (C,m,n) where C is either a groupName explained as in TorsionSubcomplex, m is the dimension of the reduced torsion subcomplex, and n is the highest vertical degree in the spectral sequence page. At the moment, the function works only when m=1,i.e, after reduction the torsion subcomplex has degree 1. It returns a component object R consists of the first page of spectral sequence, and i-th cohomology groups for i less than n.</p>

<p><strong class="button">Examples:</strong></p>

<p><a id="X832CEB637DC26E33" name="X832CEB637DC26E33"></a></p>

<h5>27.1-12 ExportHapCellcomplexToDisk</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ExportHapCellcomplexToDisk</code>( <var class="Arg">C</var>, <var class="Arg">groupName</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>It inputs a cell complex <span class="SimpleMath">C</span> which is stored as a variable in the memory, together with a user's desire name. In case, the input is a torsion cell complex then the user's desire name should be in the form "group_ptorsion" in order to use the function EquivariantSpectralSequencePage. The function will export C to the hard disk.</p>

<p><strong class="button">Examples:</strong></p>


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