chap14.txt

  
  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.
  
    Example  
     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 mutable matrix over an internal ring>, 
       <A 84 x 36 mutable matrix over an internal ring>, 
       <A 90 x 84 mutable matrix over an internal ring>, 
       <A 36 x 90 mutable 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.
  
    Example  
     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 mutable matrix over an internal ring>, 
       <A 36 x 84 mutable matrix over an internal ring>, 
       <A 84 x 90 mutable matrix over an internal ring>, 
       <A 90 x 36 mutable 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).
  
    Example  
     gap> SCLib.SearchByName("K3");
     [ [ 7648, "K3_16" ], [ 7649, "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.
  
    Example  
     gap> SCLib.SearchByName("K3");
     [ [ 7648, "K3_16" ], [ 7649, "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 modules 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).
  
    Example  
     gap> SCLib.SearchByName("K3");
     [ [ 7648, "K3_16" ], [ 7649, "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.
  
    Example  
     gap> SCLib.SearchByName("K3");
     [ [ 7648, "K3_16" ], [ 7649, "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 modules at degrees 
     [ 1 .. 5 ]>