<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayAvailableCellComplexes</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>Displays the cell complexes that are available in HAP.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VisualizeTorsionSkeleton</code>( <var class="Arg">groupName</var>, <var class="Arg">p</var> )</td><tdclass="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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">\(\Gamma\)</span>-equivariant CW-complex <span class="SimpleMath">\(X\)</span> all the cell stabilizer subgroups in <span class="SimpleMath">\(\Gamma\)</span> are finite. It returns the equivariant euler characteristic obtained by using mass formula <span class="SimpleMath">\(\sum_{\sigma}(-1)^{dim\sigma}\frac{1}{card(\Gamma_{\sigma})}\)</span></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">\(\Gamma\)</span>-equivariant CW-complex <span class="SimpleMath">\(X\)</span> on which all the cell stabilizer subgroups in <span class="SimpleMath">\(\Gamma\)</span> are finite. It returns the number of cells in <span class="SimpleMath">\(X\)</span></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">\(\Gamma\)</span>-equivariant CW-complex <span class="SimpleMath">\(X\)</span> on which all the cell stabilizer subgroups in <span class="SimpleMath">\(\Gamma\)</span> are finite. It returns the number of cells in <span class="SimpleMath">\(X\)</span> appear during the subdivision process using the RigidFacetsSubdivision.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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">\(\Gamma\)</span>-equivariant CW-complex <span class="SimpleMath">\(X\)</span> on which all the cell stabilizer subgroups in <span class="SimpleMath">\(\Gamma\)</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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
