  
  [1X27 [33X[0;0YTorsion Subcomplexes[133X[101X
  
  [33X[0;0YThe  Torsion Subcomplex subpackage has been conceived and implemented by [12XBui
  Anh Tuan[112X and [12X Alexander D. Rahm[112X.[133X
  
  
  [1X27.1 [33X[0;0Y [133X[101X
  
  [1X27.1-1 RigidFacetsSubdivision[101X
  
  [33X[1;0Y[29X[2XRigidFacetsSubdivision[102X( [3XX[103X ) [32X function[133X
  
  [33X[0;0YIt  inputs an [22Xn[122X-dimensional [22XG[122X-equivariant CW-complex [22XX[122X on which all the cell
  stabilizer   subgroups   in  [22XG[122X  are  finite.  It  returns  an  [22Xn[122X-dimensional
  [22XG[122X-equivariant  CW-complex  [22XY[122X  which  is  topologically  the  same  as [22XX[122X, but
  equipped with a [22XG[122X-CW-structure which is rigid.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-2 IsPNormal[101X
  
  [33X[1;0Y[29X[2XIsPNormal[102X( [3XG[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YInputs a finite group [22XG[122X and a prime [22Xp[122X. 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.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-3 TorsionSubcomplex[101X
  
  [33X[1;0Y[29X[2XTorsionSubcomplex[102X( [3XC[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YInputs 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:[133X
  [33X[0;0Y"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)",[133X
  [33X[0;0Ywhere  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 [22X(1+sqrt(-m))/2[122X.[133X
  [33X[0;0YThe  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.[133X
  [33X[0;0YIt is also possible to input the cell complexes[133X
  [33X[0;0Y"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)"[133X
  [33X[0;0Yprovided by [12XMathieu Dutour[112X.[133X
  
  [33X[0;0Y[12XExamples:[112X       1       ([7X../www/SideLinks/About/aboutCubical.html[107X) ,       2
  ([7X../www/SideLinks/About/aboutKnotsQuandles.html[107X) ,                         3
  ([7X../www/SideLinks/About/aboutLieCovers.html[107X) [133X
  
  [1X27.1-4 DisplayAvailableCellComplexes[101X
  
  [33X[1;0Y[29X[2XDisplayAvailableCellComplexes[102X(  ) [32X function[133X
  
  [33X[0;0YDisplays the cell complexes that are available in HAP.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-5 VisualizeTorsionSkeleton[101X
  
  [33X[1;0Y[29X[2XVisualizeTorsionSkeleton[102X( [3XgroupName[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YExecutes  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.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-6 ReduceTorsionSubcomplex[101X
  
  [33X[1;0Y[29X[2XReduceTorsionSubcomplex[102X( [3XC[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YThis   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.[133X
  [33X[0;0YIt  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.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-7 EquivariantEulerCharacteristic[101X
  
  [33X[1;0Y[29X[2XEquivariantEulerCharacteristic[102X( [3XX[103X ) [32X function[133X
  
  [33X[0;0YIt   inputs  an  [22Xn[122X-dimensional  [22XΓ[122X-equivariant  CW-complex  [22XX[122X  all  the  cell
  stabilizer  subgroups  in  [22XΓ[122X  are  finite.  It returns the equivariant euler
  characteristic obtained by using mass formula [22X∑_σ(-1)^dimσfrac1card(Γ_σ)[122X[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-8 CountingCellsOfACellComplex[101X
  
  [33X[1;0Y[29X[2XCountingCellsOfACellComplex[102X( [3XX[103X ) [32X function[133X
  
  [33X[0;0YIt  inputs an [22Xn[122X-dimensional [22XΓ[122X-equivariant CW-complex [22XX[122X on which all the cell
  stabilizer subgroups in [22XΓ[122X are finite. It returns the number of cells in [22XX[122X[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-9 CountingControlledSubdividedCells[101X
  
  [33X[1;0Y[29X[2XCountingControlledSubdividedCells[102X( [3XX[103X ) [32X function[133X
  
  [33X[0;0YIt  inputs an [22Xn[122X-dimensional [22XΓ[122X-equivariant CW-complex [22XX[122X on which all the cell
  stabilizer  subgroups  in  [22XΓ[122X are finite. It returns the number of cells in [22XX[122X
  appear during the subdivision process using the RigidFacetsSubdivision.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-10 CountingBaryCentricSubdividedCells[101X
  
  [33X[1;0Y[29X[2XCountingBaryCentricSubdividedCells[102X( [3XX[103X ) [32X function[133X
  
  [33X[0;0YIt  inputs an [22Xn[122X-dimensional [22XΓ[122X-equivariant CW-complex [22XX[122X on which all the cell
  stabilizer  subgroups  in  [22XΓ[122X are finite. It returns the number of cells in [22XX[122X
  appear during the subdivision process using the barycentric subdivision.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-11 EquivariantSpectralSequencePage[101X
  
  [33X[1;0Y[29X[2XEquivariantSpectralSequencePage[102X( [3XC[103X, [3Xm[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YIt  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.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X27.1-12 ExportHapCellcomplexToDisk[101X
  
  [33X[1;0Y[29X[2XExportHapCellcomplexToDisk[102X( [3XC[103X, [3XgroupName[103X ) [32X function[133X
  
  [33X[0;0YIt  inputs  a  cell  complex  [22XC[122X 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.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
