GAP (simpcomp) - Chapter 14: Interfaces to other software packages

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14 Interfaces to other software packages
14.1 Interface to the GAP-package homalg
14.1-1 SCHomalgBoundaryMatrices
14.1-2 SCHomalgCoboundaryMatrices
14.1-3 SCHomalgHomology
14.1-4 SCHomalgHomologyBasis
14.1-5 SCHomalgCohomology
14.1-6 SCHomalgCohomologyBasis

### 14 Interfaces to other software packages

simpcomp contains various interfaces to other software packages (see Chapter 13 for file-related export and import formats). In this chapter, some more sophisticated interfaces to other software packages are described.

Note that this chapter is subject to change and extension as it is planned to expand simpcomp's functionality in this area in the course of the next versions.

#### 14.1 Interface to the GAP-package homalg

As of Version 1.5, simpcomp is equipped with an interface to the GAP-package homalg [BR08] by Mohamed Barakat. This allows to use homalg's powerful capabilities in the field of homological algebra to compute topological properties of simplicial complexes.

For the time being, the only functions provided are ones allowing to compute the homology and cohomology groups of simplicial complexes with arbitrary coefficients. It is planned to extend the functionality in future releases of simpcomp. See below for a list of functions that provide an interface to homalg.

##### 14.1-1 SCHomalgBoundaryMatrices

`‣ SCHomalgBoundaryMatrices`( complex, modulus ) | ( method )

Returns: a list of homalg objects upon success, `fail` otherwise.

This function computes the boundary operator matrices for the simplicial complex complex with a ring of coefficients as specified by modulus: a value of `0` yields Q-matrices, a value of `1` yields Z-matrices and a value of `q`, q a prime or a prime power, computes the F_q-matrices.

     gap> SCLib.SearchByName("CP^2 (VT)");
     [ [ 16, "CP^2 (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCHomalgBoundaryMatrices(c,0);
     [ <A 36 x 9 matrix over an internal ring>,
       <A 84 x 36 matrix over an internal ring>,
       <A 90 x 84 matrix over an internal ring>,
       <A 36 x 90 matrix over an internal ring>,
       <An unevaluated 0 x 36 zero matrix over an internal ring> ]

##### 14.1-2 SCHomalgCoboundaryMatrices

`‣ SCHomalgCoboundaryMatrices`( complex, modulus ) | ( method )

Returns: a list of homalg objects upon success, `fail` otherwise.

This function computes the coboundary operator matrices for the simplicial complex complex with a ring of coefficients as specified by modulus: a value of `0` yields Q-matrices, a value of `1` yields Z-matrices and a value of `q`, q a prime or a prime power, computes the F_q-matrices.

     gap> SCLib.SearchByName("CP^2 (VT)");
     [ [ 16, "CP^2 (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCHomalgCoboundaryMatrices(c,0);
     [ <A 9 x 36 matrix over an internal ring>,
       <A 36 x 84 matrix over an internal ring>,
       <A 84 x 90 matrix over an internal ring>,
       <A 90 x 36 matrix over an internal ring>,
       <An unevaluated 36 x 0 zero matrix over an internal ring> ]

##### 14.1-3 SCHomalgHomology

`‣ SCHomalgHomology`( complex, modulus ) | ( method )

Returns: a list of integers upon success, `fail` otherwise.

This function computes the ranks of the homology groups of complex with a ring of coefficients as specified by modulus: a value of `0` computes the Q-homology, a value of `1` computes the Z-homology and a value of `q`, q a prime or a prime power, computes the F_q-homology ranks.

Note that if you are interested not only in the ranks of the homology groups, but rather their full structure, have a look at the function `SCHomalgHomologyBasis` (14.1-4).

     gap> SCLib.SearchByName("K3");
     [ [ 520, "K3_16" ], [ 539, "K3_17" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCHomalgHomology(c,0);
     #I  SCHomalgHomologyOp: Q-homology ranks: [ 1, 0, 22, 0, 1 ]
     [ 1, 0, 22, 0, 1 ]

##### 14.1-4 SCHomalgHomologyBasis

`‣ SCHomalgHomologyBasis`( complex, modulus ) | ( method )

Returns: a homalg object upon success, `fail` otherwise.

This function computes the homology groups (including explicit bases of the modules involved) of complex with a ring of coefficients as specified by modulus: a value of `0` computes the Q-homology, a value of `1` computes the Z-homology and a value of `q`, q a prime or a prime power, computes the F_q-homology groups.

The k-th homology group `hk` can be obtained by calling `hk:=CertainObject(homology,k);`, where `homology` is the homalg object returned by this function. The generators of `hk` can then be obtained via `GeneratorsOfModule(hk);`.

Note that if you are only interested in the ranks of the homology groups, then it is better to use the funtion `SCHomalgHomology` (14.1-3) which is way faster.

     gap> SCLib.SearchByName("K3");
     [ [ 520, "K3_16" ], [ 539, "K3_17" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCHomalgHomologyBasis(c,0);
     #I  SCHomalgHomologyBasisOp: constructed Q-homology groups.
     <A graded homology object consisting of 5 left vector spaces at degrees
     [ 0 .. 4 ]>

##### 14.1-5 SCHomalgCohomology

`‣ SCHomalgCohomology`( complex, modulus ) | ( method )

Returns: a list of integers upon success, `fail` otherwise.

This function computes the ranks of the cohomology groups of complex with a ring of coefficients as specified by modulus: a value of `0` computes the Q-cohomology, a value of `1` computes the Z-cohomology and a value of `q`, q a prime or a prime power, computes the F_q-cohomology ranks.

Note that if you are interested not only in the ranks of the cohomology groups, but rather their full structure, have a look at the function `SCHomalgCohomologyBasis` (14.1-6).

     gap> SCLib.SearchByName("K3");
     [ [ 520, "K3_16" ], [ 539, "K3_17" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCHomalgCohomology(c,0);
     #I  SCHomalgCohomologyOp: Q-cohomology ranks: [ 1, 0, 22, 0, 1 ]
     [ 1, 0, 22, 0, 1 ]

##### 14.1-6 SCHomalgCohomologyBasis

`‣ SCHomalgCohomologyBasis`( complex, modulus ) | ( method )

Returns: a homalg object upon success, `fail` otherwise.

This function computes the cohomology groups (including explicit bases of the modules involved) of complex with a ring of coefficients as specified by modulus: a value of `0` computes the Q-cohomology, a value of `1` computes the Z-cohomology and a value of `q`, q a prime or a prime power, computes the F_q-homology groups.

The k-th cohomology group `ck` can be obtained by calling `ck:=CertainObject(cohomology,k);`, where `cohomology` is the homalg object returned by this function. The generators of `ck` can then be obtained via `GeneratorsOfModule(ck);`.

Note that if you are only interested in the ranks of the cohomology groups, then it is better to use the funtion `SCHomalgCohomology` (14.1-5) which is way faster.

     gap> SCLib.SearchByName("K3");
     [ [ 520, "K3_16" ], [ 539, "K3_17" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCHomalgCohomologyBasis(c,0);
     #I  SCHomalgCohomologyBasisOp: constructed Q-cohomology groups.
     <A graded cohomology object consisting of 5 left vector spaces at degrees
     [ 1 .. 5 ]>

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