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<Chapter><Heading>Torsion Subcomplexes</Heading> The Torsion Subcomplex subpackage has been conceived and implemented by <B>Bui Anh Tuan</B> and <B> Alexander D. Rahm</B>. <Section><Heading> </Heading>
<ManSection> <Func Name="RigidFacetsSubdivision" Arg=" X "/> <Description> <P/> It inputs an <M>n</M>-dimensional <M>G</M>-equivariant CW-complex <M>X</M> on which all the cell stabilizer subgroups in <M>G</M> are finite. It returns an <M>n</M>-dimensional <M>G</M>-equivariant CW-complex <M>Y</M> which is topologically the same as <M>X</M>, but equipped with a <M>G</M>-CW-structure which is rigid. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="IsPNormal" Arg=" G, p"/> <Description> <P/> Inputs a finite group <M>G</M> and a prime <M>p</M>. 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/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="TorsionSubcomplex" Arg=" C, p"/> <Description> <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 <M>(1+sqrt(-m))/2</M>. <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 <B>Mathieu Dutour</B>. <P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutCubical.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutKnotsQuandles.html</Link><LinkText>2</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutLieCovers.html</Link><LinkText>3</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="DisplayAvailableCellComplexes" Arg=""/> <Description> <P/> Displays the cell complexes that are available in HAP. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="VisualizeTorsionSkeleton" Arg=" groupName, p"/> <Description> <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/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="ReduceTorsionSubcomplex" Arg=" C, p"/> <Description> <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/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="EquivariantEulerCharacteristic" Arg=" X "/> <Description> <P/> It inputs an <M>n</M>-dimensional <M>\Gamma</M>-equivariant CW-complex <M>X</M> all the cell stabilizer subgroups in <M>\Gamma</M> are finite. It returns the equivariant euler characteristic obtained by using mass formula <M>\sum_{\sigma}(-1)^{dim\sigma}\frac{1}{card(\Gamma_{\sigma})}</M> <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="CountingCellsOfACellComplex" Arg=" X "/> <Description> <P/> It inputs an <M>n</M>-dimensional <M>\Gamma</M>-equivariant CW-complex <M>X</M> on which all the cell stabilizer subgroups in <M>\Gamma</M> are finite. It returns the number of cells in <M>X</M> <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="CountingControlledSubdividedCells" Arg=" X "/> <Description> <P/> It inputs an <M>n</M>-dimensional <M>\Gamma</M>-equivariant CW-complex <M>X</M> on which all the cell stabilizer subgroups in <M>\Gamma</M> are finite. It returns the number of cells in <M>X</M> appear during the subdivision process using the RigidFacetsSubdivision. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="CountingBaryCentricSubdividedCells" Arg=" X "/> <Description> <P/> It inputs an <M>n</M>-dimensional <M>\Gamma</M>-equivariant CW-complex <M>X</M> on which all the cell stabilizer subgroups in <M>\Gamma</M> are finite. It returns the number of cells in <M>X</M> appear during the subdivision process using the barycentric subdivision. <P/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="EquivariantSpectralSequencePage" Arg=" C, m, n"/> <Description> <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/><B>Examples:</B>
</Description> </ManSection>
<ManSection> <Func Name="ExportHapCellcomplexToDisk" Arg=" C, groupName"/> <Description> <P/> It inputs a cell complex <M>C</M> 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/><B>Examples:</B>
</Description> </ManSection> </Section> </Chapter>
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