chap8.txt

  
  8 (Co-)Homology of simplicial complexes
  
  By  default,  simpcomp uses an algorithm based on discrete Morse theory (see
  Chapter 12, SCHomology (7.3-9)) for its homology computations. However, some
  additional (co-)homology related functionality cannot be realised using this
  algorithm. For this, simpcomp contains an additional (co-)homology algorithm
  (cf. SCHomologyInternal (8.1-5)), which will be presented in this chapter.
  
  Furthermore,  whenever  possible  simpcomp  makes  use  of  the  GAP package
  ''homology''  [DHSW11],  for  an  alternative  method  to calculate homology
  groups  (cf. SCHomologyClassic (6.9-31)) which sometimes is much faster than
  the built-in discrete Morse theory algorithm.
  
  
  8.1 Homology computation
  
  Apart  from  calculating  boundaries  of simplices, boundary matrices or the
  simplicial  homology  of a given complex, simpcomp is also able to compute a
  basis of the homology groups.
  
  8.1-1 SCBoundaryOperatorMatrix
  
  SCBoundaryOperatorMatrix( complex, k )  method
  Returns:  a rectangular matrix upon success, fail otherwise.
  
  Calculates the matrix of the boundary operator ∂_k+1 of a simplicial complex
  complex. Note that each column contains the boundaries of a k+1-simplex as a
  list  of oriented k-simplices and that the matrix is stored as a list of row
  vectors (as usual in GAP).
  
    Example  
     gap> c:=SCFromFacets([[1,2,3],[1,2,6],[1,3,5],[1,4,5],[1,4,6],\
                           [2,3,4],[2,4,5],[2,5,6],[3,4,6],[3,5,6]]);;
     gap> mat:=SCBoundaryOperatorMatrix(c,1);
     [ [ 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], 
       [ -1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0 ], 
       [ 0, -1, 0, 0, 0, -1, 0, 0, 0, 1, 1, 1, 0, 0, 0 ], 
       [ 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 1, 0 ], 
       [ 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, 0, 1 ], 
       [ 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, -1, 0, -1, -1 ] ]
     
  
  
  8.1-2 SCBoundarySimplex
  
  SCBoundarySimplex( simplex, orientation )  function
  Returns:  a list upon success, fail otherwise.
  
  Calculates  the  boundary of a given simplex. If the flag orientation is set
  to  true,  the function returns the boundary as a list of oriented simplices
  of  the  form [ ORIENTATION, SIMPLEX ], where ORIENTATION is either +1 or -1
  and  a  value of +1 means that SIMPLEX is positively oriented and a value of
  -1  that  SIMPLEX is negatively oriented. If orientation is set to false, an
  unoriented list of simplices is returned.
  
    Example  
     gap> SCBoundarySimplex([1..5],true);
     [ [ -1, [ 2, 3, 4, 5 ] ], [ 1, [ 1, 3, 4, 5 ] ], [ -1, [ 1, 2, 4, 5 ] ], 
       [ 1, [ 1, 2, 3, 5 ] ], [ -1, [ 1, 2, 3, 4 ] ] ]
     gap> SCBoundarySimplex([1..5],false);
     [ [ 2, 3, 4, 5 ], [ 1, 3, 4, 5 ], [ 1, 2, 4, 5 ], [ 1, 2, 3, 5 ], 
       [ 1, 2, 3, 4 ] ]
     
  
  
  8.1-3 SCHomologyBasis
  
  SCHomologyBasis( complex, k )  method
  Returns:  a list of pairs of the form [ integer, list of linear combinations
            of simplices ] upon success, fail otherwise.
  
  Calculates  a  set  of  basis  elements for the k-dimensional homology group
  (with  integer coefficients) of a simplicial complex complex. The entries of
  the  returned  list are of the form [ MODULUS, [ BASEELM1, BASEELM2, ...] ],
  where  the value MODULUS is 1 for the basis elements of the free part of the
  k-th  homology  group and q≥ 2 for the basis elements of the q-torsion part.
  In  contrast  to  the  function SCHomologyBasisAsSimplices (8.1-4) the basis
  elements  are  stored  as  lists of coefficient-index pairs referring to the
  simplices  of  the  complex,  i.e.  a basis element of the form [ [ λ_1, i],
  [λ_2,  j],  dots  ]  dots encodes the linear combination of simplices of the
  form λ_1*∆_1+λ_2*∆_2 with ∆_1=SCSkel(complex,k)[i], ∆_2=SCSkel(complex,k)[j]
  and so on.
  
    Example  
     gap> SCLib.SearchByName("(S^2xS^1)#RP^3");
     [ [ 590, "(S^2xS^1)#RP^3" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCHomologyBasis(c,1);
     [ [ 1, [ [ 1, 12 ], [ -1, 7 ], [ 1, 1 ] ] ], 
       [ 2, [ [ 1, 68 ], [ -1, 69 ], [ -1, 71 ], [ 2, 72 ], [ -2, 73 ] ] ] ]
     
  
  
  8.1-4 SCHomologyBasisAsSimplices
  
  SCHomologyBasisAsSimplices( complex, k )  method
  Returns:  a list of pairs of the form [ integer, list of linear combinations
            of simplices ] upon success, fail otherwise.
  
  Calculates  a  set  of  basis  elements for the k-dimensional homology group
  (with  integer coefficients) of a simplicial complex complex. The entries of
  the  returned  list are of the form [ MODULUS, [ BASEELM1, BASEELM2, ...] ],
  where  the value MODULUS is 1 for the basis elements of the free part of the
  k-th  homology  group and q≥ 2 for the basis elements of the q-torsion part.
  In  contrast  to the function SCHomologyBasis (8.1-3) the basis elements are
  stored  as  lists  of coefficient-simplex pairs, i.e. a basis element of the
  form  [  [  λ_1,  ∆_1], [λ_2, ∆_2], dots ] encodes the linear combination of
  simplices of the form λ_1*∆_1+λ_2*∆_2 + dots.
  
    Example  
     gap> SCLib.SearchByName("(S^2xS^1)#RP^3");
     [ [ 590, "(S^2xS^1)#RP^3" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCHomologyBasisAsSimplices(c,1);
     [ [ 1, [ [ 1, [ 2, 8 ] ], [ -1, [ 1, 8 ] ], [ 1, [ 1, 2 ] ] ] ], 
       [ 2, 
           [ [ 1, [ 11, 12 ] ], [ -1, [ 11, 13 ] ], [ -1, [ 12, 13 ] ], 
               [ 2, [ 12, 14 ] ], [ -2, [ 13, 14 ] ] ] ] ]
     
  
  
  8.1-5 SCHomologyInternal
  
  SCHomologyInternal( complex )  function
  Returns:  a  list  of pairs of the form [ integer, list ] upon success, fail
            otherwise.
  
  This   function  computes  the  reduced  simplicial  homology  with  integer
  coefficients   of   a   given   simplicial   complex  complex  with  integer
  coefficients. It uses the algorithm described in [DKT08].
  
  The  output is a list of homology groups of the form [H_0,....,H_d], where d
  is  the dimension of complex. The format of the homology groups H_i is given
  in terms of their maximal cyclic subgroups, i.e. a homology group H_i≅ Z^f +
  Z  /  t_1  Z  ×  dots  ×  Z  /  t_n  Z  is  returned  in form of a list [ f,
  [t_1,...,t_n]  ],  where f is the (integer) free part of H_i and t_i denotes
  the  torsion  parts  of  H_i  ordered  in  weakly  incresing  size. See also
  SCHomology (7.3-9) and SCHomologyClassic (6.9-31).
  
    Example  
     gap> c:=SCSurface(1,false);;
     gap> SCHomologyInternal(c);
     [ [ 0, [  ] ], [ 0, [ 2 ] ], [ 0, [  ] ] ]
     
  
  
  
  8.2 Cohomology computation
  
  simpcomp  can  also  compute  the cohomology groups of simplicial complexes,
  bases  of  these  cohomology groups, the cup product of two cocycles and the
  intersection form of (orientable) 4-manifolds.
  
  8.2-1 SCCoboundaryOperatorMatrix
  
  SCCoboundaryOperatorMatrix( complex, k )  method
  Returns:  a rectangular matrix upon success, fail otherwise.
  
  Calculates  the  matrix  of  the  coboundary operator d^k+1 as a list of row
  vectors.
  
    Example  
     gap> c:=SCFromFacets([[1,2,3],[1,2,6],[1,3,5],[1,4,5],[1,4,6],\
                           [2,3,4],[2,4,5],[2,5,6],[3,4,6],[3,5,6]]);
     > [SimplicialComplex
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="unnamed complex 2"
      Dim=2
     
     /SimplicialComplex]
     gap> mat:=SCCoboundaryOperatorMatrix(c,1);
     [ [ -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], 
       [ -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0 ], 
       [ 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0 ], 
       [ 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0 ], 
       [ 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0 ], 
       [ 0, 0, 0, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 0, 0 ], 
       [ 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, -1, 0, 0 ], 
       [ 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1 ], 
       [ 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, -1, 0 ], 
       [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, -1 ] ]
     
  
  
  8.2-2 SCCohomology
  
  SCCohomology( complex )  method
  Returns:  a  list  of pairs of the form [ integer, list ] upon success, fail
            otherwise.
  
  This   function  computes  the  simplicial  cohomology  groups  of  a  given
  simplicial  complex complex with integer coefficients. It uses the algorithm
  described in [DKT08].
  
  The  output is a list of cohomology groups of the form [H^0,....,H^d], where
  d  is  the  dimension of complex. The format of the cohomology groups H^i is
  given  in  terms  of their maximal cyclic subgroups, i.e. a cohomology group
  H^i≅  Z^f  + Z / t_1 Z × dots × Z / t_n Z is returned in form of a list [ f,
  [t_1,...,t_n]  ],  where f is the (integer) free part of H^i and t_i denotes
  the torsion parts of H^i ordered in weakly increasing size.
  
    Example  
     gap> c:=SCFromFacets([[1,2,3],[1,2,6],[1,3,5],[1,4,5],[1,4,6],
                           [2,3,4],[2,4,5],[2,5,6],[3,4,6],[3,5,6]]);
     > [SimplicialComplex
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="unnamed complex 4"
      Dim=2
     
     /SimplicialComplex]
     gap> SCCohomology(c);
     [ [ 1, [  ] ], [ 0, [  ] ], [ 0, [ 2 ] ] ]
     
  
  
  8.2-3 SCCohomologyBasis
  
  SCCohomologyBasis( complex, k )  method
  Returns:  a list of pairs of the form [ integer, list of linear combinations
            of simplices ] upon success, fail otherwise.
  
  Calculates  a  set  of basis elements for the k-dimensional cohomology group
  (with  integer coefficients) of a simplicial complex complex. The entries of
  the  returned  list are of the form [ MODULUS, [ BASEELM1, BASEELM2, ...] ],
  where  the value MODULUS is 1 for the basis elements of the free part of the
  k-th  homology  group and q≥ 2 for the basis elements of the q-torsion part.
  In  contrast  to the function SCCohomologyBasisAsSimplices (8.2-4) the basis
  elements  are  stored  as  lists of coefficient-index pairs referring to the
  linear  forms  dual to the simplices in the k-th cochain complex of complex,
  i.e.  a basis element of the form [ [ λ_1, i], [λ_2, j], dots ] dots encodes
  the  linear  combination  of  simplices  (or  their dual linear forms in the
  corresponding   cochain   complex)   of   the   form   λ_1*∆_1+λ_2*∆_2  with
  ∆_1=SCSkel(complex,k)[i], ∆_2=SCSkel(complex,k)[j] and so on.
  
    Example  
     gap> SCLib.SearchByName("SU(3)/SO(3)");   
     [ [ 563, "SU(3)/SO(3) (VT)" ], [ 7276, "SU(3)/SO(3) (VT)" ], 
       [ 7418, "SU(3)/SO(3) (VT)" ], [ 7419, "SU(3)/SO(3) (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCCohomologyBasis(c,3); 
     [ [ 2, [ [ -9, 259 ], [ 9, 262 ], [ 9, 263 ], [ -9, 270 ], [ 9, 271 ], 
               [ -9, 273 ], [ -9, 274 ], [ -18, 275 ], [ -9, 276 ], [ 9, 278 ], 
               [ -9, 279 ], [ -9, 280 ], [ 3, 283 ], [ -3, 285 ], [ 3, 289 ], 
               [ -3, 294 ], [ 3, 310 ], [ -3, 313 ], [ 3, 316 ], [ -1, 317 ], 
               [ -6, 318 ], [ 3, 319 ], [ -6, 320 ], [ 6, 321 ], [ 1, 322 ], 
               [ 3, 325 ], [ -1, 328 ], [ 6, 330 ], [ -2, 331 ], [ 12, 332 ], 
               [ 7, 333 ], [ -5, 334 ], [ 1, 345 ], [ 3, 355 ], [ -9, 357 ], 
               [ 9, 358 ], [ 1, 363 ], [ 12, 365 ], [ -9, 366 ], [ -3, 370 ], 
               [ -1, 371 ], [ -3, 372 ], [ 8, 373 ], [ -1, 374 ], [ 6, 375 ], 
               [ 9, 376 ], [ 3, 377 ], [ 1, 380 ], [ 3, 383 ], [ -8, 385 ], 
               [ -9, 386 ], [ -9, 388 ], [ -18, 404 ], [ 9, 410 ], [ -9, 425 ], 
               [ -18, 426 ], [ -9, 427 ], [ 9, 428 ], [ -9, 429 ], [ 3, 433 ], 
               [ -3, 435 ], [ -9, 437 ], [ 10, 442 ], [ 12, 445 ], [ 1, 447 ], 
               [ -19, 448 ], [ 2, 449 ], [ -1, 450 ], [ -9, 451 ], [ 3, 453 ], 
               [ 1, 455 ], [ 1, 457 ], [ -11, 458 ], [ -9, 459 ], [ 9, 461 ], 
               [ 9, 462 ], [ -9, 468 ], [ 9, 469 ], [ -18, 471 ], [ -9, 472 ], 
               [ 9, 474 ], [ -9, 475 ], [ 9, 488 ], [ 9, 495 ], [ -9, 500 ], 
               [ -3, 504 ], [ 9, 505 ], [ 9, 512 ], [ 9, 515 ], [ 6, 519 ], 
               [ 18, 521 ], [ -15, 523 ], [ 9, 524 ], [ -3, 525 ], [ 18, 527 ], 
               [ -18, 528 ], [ 6, 529 ], [ 6, 531 ], [ 12, 532 ] ] ] ]
     
  
  
  8.2-4 SCCohomologyBasisAsSimplices
  
  SCCohomologyBasisAsSimplices( complex, k )  method
  Returns:  a  list  of  pars  of  the  form  [ integer, linear combination of
            simplices ] upon success, fail otherwise.
  
  Calculates  a  set  of basis elements for the k-dimensional cohomology group
  (with  integer coefficients) of a simplicial complex complex. The entries of
  the  returned  list are of the form [ MODULUS, [ BASEELM1, BASEELM2, ...] ],
  where  the value MODULUS is 1 for the basis elements of the free part of the
  k-th  homology  group and q≥ 2 for the basis elements of the q-torsion part.
  In contrast to the function SCCohomologyBasis (8.2-3) the basis elements are
  stored  as  lists of coefficient-simplex pairs referring to the linear forms
  dual  to  the simplices in the k-th cochain complex of complex, i.e. a basis
  element  of  the  form  [  [  λ_1, ∆_i], [λ_2, ∆_j], dots ] dots encodes the
  linear  combination  of  simplices  (or  their  dual  linear  forms  in  the
  corresponding cochain complex) of the form λ_1*∆_1+λ_2*∆_2 + dots.
  
    Example  
     gap> SCLib.SearchByName("SU(3)/SO(3)");   
     [ [ 563, "SU(3)/SO(3) (VT)" ], [ 7276, "SU(3)/SO(3) (VT)" ], 
       [ 7418, "SU(3)/SO(3) (VT)" ], [ 7419, "SU(3)/SO(3) (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCCohomologyBasisAsSimplices(c,3);
     [ [ 2, 
           [ [ -9, [ 2, 7, 8, 9 ] ], [ 9, [ 2, 7, 8, 12 ] ], 
               [ 9, [ 2, 7, 8, 13 ] ], [ -9, [ 2, 7, 11, 12 ] ], 
               [ 9, [ 2, 7, 11, 13 ] ], [ -9, [ 2, 8, 9, 10 ] ], 
               [ -9, [ 2, 8, 9, 11 ] ], [ -18, [ 2, 8, 9, 12 ] ], 
               [ -9, [ 2, 8, 9, 13 ] ], [ 9, [ 2, 8, 10, 12 ] ], 
               [ -9, [ 2, 8, 10, 13 ] ], [ -9, [ 2, 8, 11, 12 ] ], 
               [ 3, [ 2, 9, 10, 12 ] ], [ -3, [ 2, 9, 11, 12 ] ], 
               [ 3, [ 3, 4, 5, 7 ] ], [ -3, [ 3, 4, 5, 12 ] ], 
               [ 3, [ 3, 4, 10, 12 ] ], [ -3, [ 3, 5, 6, 7 ] ], 
               [ 3, [ 3, 5, 6, 11 ] ], [ -1, [ 3, 5, 6, 13 ] ], 
               [ -6, [ 3, 5, 7, 8 ] ], [ 3, [ 3, 5, 7, 10 ] ], 
               [ -6, [ 3, 5, 7, 11 ] ], [ 6, [ 3, 5, 7, 12 ] ], 
               [ 1, [ 3, 5, 7, 13 ] ], [ 3, [ 3, 5, 8, 12 ] ], 
               [ -1, [ 3, 5, 9, 13 ] ], [ 6, [ 3, 5, 10, 12 ] ], 
               [ -2, [ 3, 5, 10, 13 ] ], [ 12, [ 3, 5, 11, 12 ] ], 
               [ 7, [ 3, 5, 11, 13 ] ], [ -5, [ 3, 5, 12, 13 ] ], 
               [ 1, [ 3, 6, 9, 13 ] ], [ 3, [ 3, 7, 10, 12 ] ], 
               [ -9, [ 3, 7, 11, 12 ] ], [ 9, [ 3, 7, 11, 13 ] ], 
               [ 1, [ 3, 8, 9, 13 ] ], [ 12, [ 3, 8, 10, 12 ] ], 
               [ -9, [ 3, 8, 10, 13 ] ], [ -3, [ 3, 9, 10, 12 ] ], 
               [ -1, [ 3, 9, 10, 13 ] ], [ -3, [ 3, 9, 11, 12 ] ], 
               [ 8, [ 3, 9, 11, 13 ] ], [ -1, [ 3, 9, 12, 13 ] ], 
               [ 6, [ 3, 10, 11, 12 ] ], [ 9, [ 3, 10, 11, 13 ] ], 
               [ 3, [ 3, 10, 12, 13 ] ], [ 1, [ 4, 5, 6, 8 ] ], 
               [ 3, [ 4, 5, 6, 11 ] ], [ -8, [ 4, 5, 6, 13 ] ], 
               [ -9, [ 4, 5, 7, 8 ] ], [ -9, [ 4, 5, 7, 11 ] ], 
               [ -18, [ 4, 6, 8, 9 ] ], [ 9, [ 4, 6, 9, 13 ] ], 
               [ -9, [ 4, 8, 9, 10 ] ], [ -18, [ 4, 8, 9, 12 ] ], 
               [ -9, [ 4, 8, 9, 13 ] ], [ 9, [ 4, 8, 10, 12 ] ], 
               [ -9, [ 4, 8, 10, 13 ] ], [ 3, [ 4, 9, 10, 12 ] ], 
               [ -3, [ 4, 9, 11, 12 ] ], [ -9, [ 4, 9, 12, 13 ] ], 
               [ 10, [ 5, 6, 7, 8 ] ], [ 12, [ 5, 6, 7, 11 ] ], 
               [ 1, [ 5, 6, 7, 13 ] ], [ -19, [ 5, 6, 8, 9 ] ], 
               [ 2, [ 5, 6, 8, 11 ] ], [ -1, [ 5, 6, 8, 12 ] ], 
               [ -9, [ 5, 6, 8, 13 ] ], [ 3, [ 5, 6, 9, 11 ] ], 
               [ 1, [ 5, 6, 9, 13 ] ], [ 1, [ 5, 6, 10, 13 ] ], 
               [ -11, [ 5, 6, 11, 13 ] ], [ -9, [ 5, 7, 8, 9 ] ], 
               [ 9, [ 5, 7, 8, 12 ] ], [ 9, [ 5, 7, 8, 13 ] ], 
               [ -9, [ 5, 7, 11, 12 ] ], [ 9, [ 5, 7, 11, 13 ] ], 
               [ -18, [ 5, 8, 9, 12 ] ], [ -9, [ 5, 8, 9, 13 ] ], 
               [ 9, [ 5, 8, 10, 12 ] ], [ -9, [ 5, 8, 11, 12 ] ], 
               [ 9, [ 6, 7, 8, 13 ] ], [ 9, [ 6, 7, 11, 13 ] ], 
               [ -9, [ 6, 8, 10, 13 ] ], [ -3, [ 6, 9, 11, 12 ] ], 
               [ 9, [ 6, 9, 11, 13 ] ], [ 9, [ 7, 8, 9, 13 ] ], 
               [ 9, [ 7, 8, 11, 12 ] ], [ 6, [ 7, 9, 11, 12 ] ], 
               [ 18, [ 7, 11, 12, 13 ] ], [ -15, [ 8, 9, 10, 12 ] ], 
               [ 9, [ 8, 9, 10, 13 ] ], [ -3, [ 8, 9, 11, 12 ] ], 
               [ 18, [ 8, 10, 11, 12 ] ], [ -18, [ 8, 10, 12, 13 ] ], 
               [ 6, [ 9, 10, 11, 12 ] ], [ 6, [ 9, 10, 12, 13 ] ], 
               [ 12, [ 9, 11, 12, 13 ] ] ] ] ]
     
  
  
  8.2-5 SCCupProduct
  
  SCCupProduct( complex, cocycle1, cocycle2 )  function
  Returns:  a list of pairs of the form [ ORIENTATION, SIMPLEX ] upon success,
            fail otherwise.
  
  The  cup  product is a method of adjoining two cocycles of degree p and q to
  form a composite cocycle of degree p + q. It endows the cohomology groups of
  a simplicial complex with the structure of a ring.
  
  The  construction  of  the cup product starts with a product of cochains: if
  cocycle1  is a p-cochain and cocylce2 is a q-cochain of a simplicial complex
  complex (given as list of oriented p- (q-)simplices), then
  
  cocycle1  ⌣  cocycle2(σ) =cocycle1(σ ∘ ι_0,1, ... p) ⋅ cocycle2(σ ∘ ι_p, p+1
  ,..., p + q)
  
  where  σ  is  a  p  + q-simplex and ι_S, S ⊂ {0,1,...,p+q } is the canonical
  embedding of the simplex spanned by S into the (p + q)-standard simplex.
  
  σ ∘ ι_0,1, ..., p is called the p-th front face and σ ∘ ι_p, p+1, ..., p + q
  is the q-th back face of σ, respectively.
  
  Note  that  this  function  only  computes  the cup product in the case that
  complex  is an orientable weak pseudomanifold of dimension 2k and p = q = k.
  Furthermore,  complex  must be given in standard labeling, with sorted facet
  list and cocylce1 and cocylce2 must be given in simplex notation and labeled
  accordingly. Note that the latter condition is usually fulfilled in case the
  cocycles were computed using SCCohomologyBasisAsSimplices (8.2-4).
  
    Example  
     gap> SCLib.SearchByName("K3");
     [ [ 7648, "K3_16" ], [ 7649, "K3_17" ] ]
     gap> c:=SCLib.Load(last[1][1]);;                                     
     gap> basis:=SCCohomologyBasisAsSimplices(c,2);;
     gap> SCCupProduct(c,basis[1][2],basis[1][2]);
     [ [ 1, [ 1, 2, 4, 7, 11 ] ], [ 1, [ 2, 3, 4, 5, 9 ] ] ]
     gap> SCCupProduct(c,basis[1][2],basis[2][2]);
     [ [ -1, [ 1, 2, 4, 7, 11 ] ], [ -1, [ 1, 2, 4, 7, 15 ] ], 
       [ -1, [ 2, 3, 4, 5, 9 ] ] ]
     
  
  
  8.2-6 SCIntersectionForm
  
  SCIntersectionForm( complex )  method
  Returns:  a square matrix of integer values upon success, fail otherwise.
  
  For  2k-dimensional orientable manifolds M the cup product (see SCCupProduct
  (8.2-5)) defines a bilinear form
  
  H^k ( M ) ×H^k ( M ) ->H^2k (M), (a,b) ↦ a ∪ b
  
  called  the  intersection  form of M. This function returns the intersection
  form  of an orientable combinatorial 2k-manifold complex in form of a matrix
  mat   with   respect   to   the   basis   of   H^k  (complexM)  computed  by
  SCCohomologyBasisAsSimplices  (8.2-4). The matrix entry mat[i][j] equals the
  intersection  number  of the i-th base element with the j-th base element of
  H^k (complexM).
  
    Example  
     gap> SCLib.SearchByName("CP^2");       
     [ [ 16, "CP^2 (VT)" ], [ 99, "CP^2#-CP^2" ], [ 100, "CP^2#CP^2" ], 
       [ 400, "CP^2#(S^2xS^2)" ], [ 2486, "Gaifullin CP^2" ], 
       [ 4401, "(S^3~S^1)#(CP^2)^{#5} (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);; 
     gap> c1:=SCConnectedSum(c,c);;
     gap> c2:=SCConnectedSumMinus(c,c);;
     gap> q1:=SCIntersectionForm(c1);;
     gap> q2:=SCIntersectionForm(c2);;
     gap> PrintArray(q1);
     [ [  1,  0 ],
       [  0,  1 ] ]
     gap> PrintArray(q2);
     [ [   1,   0 ],
       [   0,  -1 ] ]
     
  
  
  8.2-7 SCIntersectionFormParity
  
  SCIntersectionFormParity( complex )  method
  Returns:  0 or 1 upon success, fail otherwise.
  
  Computes  the  parity  of  the intersection form of a combinatorial manifold
  complex  (see  SCIntersectionForm  (8.2-6)). If the intersection for is even
  (i.  e. all diagonal entries are even numbers) 0 is returned, otherwise 1 is
  returned.
  
    Example  
     gap> SCLib.SearchByName("S^2xS^2");
     [ [ 59, "S^2xS^2" ], [ 134, "S^2xS^2 (VT)" ], [ 135, "S^2xS^2 (VT)" ], 
       [ 136, "S^2xS^2 (VT)" ], [ 137, "(S^2xS^2)#(S^2xS^2)" ], 
       [ 360, "(S^2xS^2)#(S^2xS^2) (VT)" ], [ 361, "(S^2xS^2)#(S^2xS^2) (VT)" ], 
       [ 400, "CP^2#(S^2xS^2)" ] ]
     gap> c:=SCLib.Load(last[1][1]);;    
     gap> SCIntersectionFormParity(c);
     0
     gap> SCLib.SearchByName("CP^2");       
     [ [ 16, "CP^2 (VT)" ], [ 99, "CP^2#-CP^2" ], [ 100, "CP^2#CP^2" ], 
       [ 400, "CP^2#(S^2xS^2)" ], [ 2486, "Gaifullin CP^2" ], 
       [ 4401, "(S^3~S^1)#(CP^2)^{#5} (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);; 
     gap> SCIntersectionFormParity(c);
     1
     
  
  
  8.2-8 SCIntersectionFormDimensionality
  
  SCIntersectionFormDimensionality( complex )  method
  Returns:  an integer upon success, fail otherwise.
  
  Returns  the  dimensionality  of  the  intersection  form of a combinatorial
  manifold  complex,  i.  e. the length of a minimal generating set of H^k (M)
  (where  2k  is the dimension of complex). See SCIntersectionForm (8.2-6) for
  further details.
  
    Example  
     gap> SCLib.SearchByName("CP^2");       
     [ [ 16, "CP^2 (VT)" ], [ 99, "CP^2#-CP^2" ], [ 100, "CP^2#CP^2" ], 
       [ 400, "CP^2#(S^2xS^2)" ], [ 2486, "Gaifullin CP^2" ], 
       [ 4401, "(S^3~S^1)#(CP^2)^{#5} (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);; 
     gap> SCIntersectionFormParity(c);
     1
     gap> SCCohomology(c);
     [ [ 1, [  ] ], [ 0, [  ] ], [ 1, [  ] ], [ 0, [  ] ], [ 1, [  ] ] ]
     gap> SCIntersectionFormDimensionality(c);
     1
     gap> d:=SCConnectedProduct(c,10);;
     gap> SCIntersectionFormDimensionality(d);
     10
     
  
  
  8.2-9 SCIntersectionFormSignature
  
  SCIntersectionFormSignature( complex )  method
  Returns:  a triple of integers upon success, fail otherwise.
  
  Computes  the  dimensionality (see SCIntersectionFormDimensionality (8.2-8))
  and  the  signature  of  the  intersection  form of a combinatorial manifold
  complex as a 3-tuple that contains the dimensionality in the first entry and
  the number of positive / negative eigenvalues in the second and third entry.
  See SCIntersectionForm (8.2-6) for further details.
  
  Internally  calls the GAP-functions Matrix_CharacteristicPolynomialSameField
  and  CoefficientsOfLaurentPolynomial  to  compute  the  number of positive /
  negative eigenvalues of the intersection form.
  
    Example  
     gap> SCLib.SearchByName("CP^2");       
     [ [ 16, "CP^2 (VT)" ], [ 99, "CP^2#-CP^2" ], [ 100, "CP^2#CP^2" ], 
       [ 400, "CP^2#(S^2xS^2)" ], [ 2486, "Gaifullin CP^2" ], 
       [ 4401, "(S^3~S^1)#(CP^2)^{#5} (VT)" ] ]
     gap> c:=SCLib.Load(last[1][1]);;
     gap> SCIntersectionFormParity(c);
     1
     gap> SCCohomology(c);
     [ [ 1, [  ] ], [ 0, [  ] ], [ 1, [  ] ], [ 0, [  ] ], [ 1, [  ] ] ]
     gap> SCIntersectionFormSignature(c);
     [ 1, 0, 1 ]
     gap> d:=SCConnectedSum(c,c);                           
     [SimplicialComplex
     
      Properties known: Dim, FacetsEx, Name, Vertices.
     
      Name="CP^2 (VT)#+-CP^2 (VT)"
      Dim=4
     
     /SimplicialComplex]
     gap> SCIntersectionFormSignature(d);
     [ 2, 2, 0 ]
     gap> d:=SCConnectedSumMinus(c,c);;
     gap> SCIntersectionFormSignature(d);
     [ 2, 1, 1 ]