GAP (HAP commands) - Chapter 27: Torsion Subcomplexes

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27 Torsion Subcomplexes
27.1
27.1-1 RigidFacetsSubdivision
27.1-2 IsPNormal
27.1-3 TorsionSubcomplex
27.1-4 DisplayAvailableCellComplexes
27.1-5 VisualizeTorsionSkeleton
27.1-6 ReduceTorsionSubcomplex
27.1-7 EquivariantEulerCharacteristic
27.1-8 CountingCellsOfACellComplex
27.1-9 CountingControlledSubdividedCells
27.1-10 CountingBaryCentricSubdividedCells
27.1-11 EquivariantSpectralSequencePage
27.1-12 ExportHapCellcomplexToDisk

### 27 Torsion Subcomplexes

The Torsion Subcomplex subpackage has been conceived and implemented by Bui Anh Tuan and Alexander D. Rahm.

#### 27.1

##### 27.1-1 RigidFacetsSubdivision

`‣ RigidFacetsSubdivision`( X ) | ( function )

It inputs an \(n\)-dimensional \(G\)-equivariant CW-complex \(X\) on which all the cell stabilizer subgroups in \(G\) are finite. It returns an \(n\)-dimensional \(G\)-equivariant CW-complex \(Y\) which is topologically the same as \(X\), but equipped with a \(G\)-CW-structure which is rigid.

Examples:

##### 27.1-2 IsPNormal

`‣ IsPNormal`( G, p ) | ( function )

Inputs a finite group \(G\) and a prime \(p\). 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.

Examples:

##### 27.1-3 TorsionSubcomplex

`‣ TorsionSubcomplex`( C, p ) | ( function )

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:
"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)",
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 \((1+sqrt(-m))/2\).
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.
It is also possible to input the cell complexes
"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)"
provided by Mathieu Dutour.

Examples: 1 , 2 , 3

##### 27.1-4 DisplayAvailableCellComplexes

`‣ DisplayAvailableCellComplexes`( ) | ( function )

Displays the cell complexes that are available in HAP.

Examples:

##### 27.1-5 VisualizeTorsionSkeleton

`‣ VisualizeTorsionSkeleton`( groupName, p ) | ( function )

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.

Examples:

##### 27.1-6 ReduceTorsionSubcomplex

`‣ ReduceTorsionSubcomplex`( C, p ) | ( function )

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.
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.

Examples:

##### 27.1-7 EquivariantEulerCharacteristic

`‣ EquivariantEulerCharacteristic`( X ) | ( function )

It inputs an \(n\)-dimensional \(\Gamma\)-equivariant CW-complex \(X\) all the cell stabilizer subgroups in \(\Gamma\) are finite. It returns the equivariant euler characteristic obtained by using mass formula \(\sum_{\sigma}(-1)^{dim\sigma}\frac{1}{card(\Gamma_{\sigma})}\)

Examples:

##### 27.1-8 CountingCellsOfACellComplex

`‣ CountingCellsOfACellComplex`( X ) | ( function )

It inputs an \(n\)-dimensional \(\Gamma\)-equivariant CW-complex \(X\) on which all the cell stabilizer subgroups in \(\Gamma\) are finite. It returns the number of cells in \(X\)

Examples:

##### 27.1-9 CountingControlledSubdividedCells

`‣ CountingControlledSubdividedCells`( X ) | ( function )

It inputs an \(n\)-dimensional \(\Gamma\)-equivariant CW-complex \(X\) on which all the cell stabilizer subgroups in \(\Gamma\) are finite. It returns the number of cells in \(X\) appear during the subdivision process using the RigidFacetsSubdivision.

Examples:

##### 27.1-10 CountingBaryCentricSubdividedCells

`‣ CountingBaryCentricSubdividedCells`( X ) | ( function )

It inputs an \(n\)-dimensional \(\Gamma\)-equivariant CW-complex \(X\) on which all the cell stabilizer subgroups in \(\Gamma\) are finite. It returns the number of cells in \(X\) appear during the subdivision process using the barycentric subdivision.

Examples:

##### 27.1-11 EquivariantSpectralSequencePage

`‣ EquivariantSpectralSequencePage`( C, m, n ) | ( function )

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.

Examples:

##### 27.1-12 ExportHapCellcomplexToDisk

`‣ ExportHapCellcomplexToDisk`( C, groupName ) | ( function )

It inputs a cell complex \(C\) 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.

Examples:

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