  
  [1X27 [33X[0;0YSimplicial Complexes[133X[101X
  
  [33X[0;0Y[10XHomology(T,n)[110X[133X
  [33X[0;0Y[10XHomology(T)[110X[133X
  
  [33X[0;0YInputs  a  pure cubical complex, or cubical complex, or simplicial complex [22XT[122X
  and  a non-negative integer [22Xn[122X. It returns the n-th integral homology of [22XT[122X as
  a  list  of torsion integers. If no value of [22Xn[122X is input then the list of all
  homologies of [22XT[122X in dimensions 0 to Dimension(T) is returned .[133X
  [33X[0;0Y[10XRipsHomology(G,n)[110X[133X
  [33X[0;0Y[10XRipsHomology(G,n,p)[110X[133X
  
  [33X[0;0YInputs  a  graph  [22XG[122X, a non-negative integer [22Xn[122X (and optionally a prime number
  [22Xp[122X).  It returns the integral homology (or mod p homology) in degree [22Xn[122X of the
  Rips complex of [22XG[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10X Bettinumbers(T,n)[110X[133X
  [33X[0;0Y[10X Bettinumbers(T)[110X[133X
  
  [33X[0;0YInputs  a  pure  cubical  complex, or cubical complex, simplicial complex or
  chain  complex [22XT[122X and a non-negative integer [22Xn[122X. The rank of the n-th rational
  homology  group  [22XH_n(T,  Q)[122X is returned. If no value for n is input then the
  list of Betti numbers in dimensions 0 to Dimension(T) is returned .[133X
  [33X[0;0Y[10XChainComplex(T)[110X[133X
  
  [33X[0;0YInputs  a  pure cubical complex, or cubical complex, or simplicial complex [22XT[122X
  and returns the (often very large) cellular chain complex of [22XT[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XCechComplexOfPureCubicalComplex(T)[110X[133X
  
  [33X[0;0YInputs  a  d-dimensional  pure  cubical  complex  [22XT[122X and returns a simplicial
  complex [22XS[122X. The simplicial complex [22XS[122X has one vertex for each d-cube in [22XT[122X, and
  an  n-simplex  for  each  collection  of n+1 d-cubes with non-trivial common
  intersection. The homotopy types of [22XT[122X and [22XS[122X are equal.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XPureComplexToSimplicialComplex(T,k)[110X[133X
  
  [33X[0;0YInputs either a d-dimensional pure cubical complex [22XT[122X or a d-dimensional pure
  permutahedral  complex  [22XT[122X together with a non-negative integer [22Xk[122X. It returns
  the  first  [22Xk[122X dimensions of a simplicial complex [22XS[122X. The simplicial complex [22XS[122X
  has one vertex for each d-cell in [22XT[122X, and an n-simplex for each collection of
  n+1  d-cells  with  non-trivial common intersection. The homotopy types of [22XT[122X
  and [22XS[122X are equal.[133X
  
  [33X[0;0YFor a pure cubical complex [22XT[122X this uses a slightly different algorithm to the
  function   CechComplexOfPureCubicalComplex(T)   but   constructs   the  same
  simplicial complex.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XRipsChainComplex(G,n)[110X[133X
  
  [33X[0;0YInputs  a  graph  [22XG[122X  and a non-negative integer [22Xn[122X. It returns [22Xn+1[122X terms of a
  chain  complex  whose homology is that of the nerve (or Rips complex) of the
  graph in degrees up to [22Xn[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XVectorsToSymmetricMatrix(M)[110X[133X
  [33X[0;0Y[10XVectorsToSymmetricMatrix(M,distance)[110X[133X
  
  [33X[0;0YInputs a matrix [22XM[122X of rational numbers and returns a symmetric matrix [22XS[122X whose
  [22X(i,j)[122X  entry  is  the distance between the [22Xi[122X-th row and [22Xj[122X-th rows of [22XM[122X where
  distance  is  given  by  the  sum  of  the absolute values of the coordinate
  differences.[133X
  
  [33X[0;0YOptionally,  a  function  distance(v,w) can be entered as a second argument.
  This  function  has  to  return  a rational number for each pair of rational
  vectors [22Xv,w[122X of length Length(M[1]).[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XEulerCharacteristic(T)[110X[133X
  
  [33X[0;0YInputs  a  pure cubical complex, or cubical complex, or simplicial complex [22XT[122X
  and returns its Euler characteristic.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XMaximalSimplicesToSimplicialComplex(L)[110X[133X
  
  [33X[0;0YInputs a list L whose entries are lists of vertices representing the maximal
  simplices  of a simplicial complex. The simplicial complex is returned. Here
  a "vertex" is a GAP object such as an integer or a subgroup.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XSkeletonOfSimplicialComplex(S,k)[110X[133X
  
  [33X[0;0YInputs a simplicial complex [22XS[122X and a positive integer [22Xk[122X less than or equal to
  the  dimension  of  [22XS[122X.  It  returns  the  truncated [22Xk[122X-dimensional simplicial
  complex [22XS^k[122X (and leaves [22XS[122X unchanged).[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XGraphOfSimplicialComplex(S)[110X[133X
  
  [33X[0;0YInputs a simplicial complex [22XS[122X and returns the graph of [22XS[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XContractibleSubcomplexOfSimplicialComplex(S)[110X[133X
  
  [33X[0;0YInputs  a simplicial complex [22XS[122X and returns a (probably maximal) contractible
  subcomplex of [22XS[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XPathComponentsOfSimplicialComplex(S,n)[110X[133X
  
  [33X[0;0YInputs a simplicial complex [22XS[122X and a nonnegative integer [22Xn[122X. If [22Xn=0[122X the number
  of  path  components  of [22XS[122X is returned. Otherwise the n-th path component is
  returned (as a simplicial complex).[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XQuillenComplex(G)[110X[133X
  
  [33X[0;0YInputs  a  finite  group  [22XG[122X  and returns, as a simplicial complex, the order
  complex of the poset of non-trivial elementary abelian subgroups of [22XG[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XSymmetricMatrixToIncidenceMatrix(S,t)[110X[133X
  [33X[0;0Y[10XSymmetricMatrixToIncidenceMatrix(S,t,d)[110X[133X
  
  [33X[0;0YInputs  a symmetric integer matrix S and an integer t. It returns the matrix
  [22XM[122X with [22XM_ij=1[122X if [22XI_ij[122X is less than [22Xt[122X and [22XI_ij=1[122X otherwise.[133X
  
  [33X[0;0YAn  optional  integer  [22Xd[122X  can be given as a third argument. In this case the
  incidence  matrix  should  have  roughly  at  most  [22Xd[122X  entries  in  each row
  (corresponding to the $d$ smallest entries in each row of [22XS[122X).[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XIncidenceMatrixToGraph(M)[110X[133X
  
  [33X[0;0YInputs  a  symmetric  0/1 matrix M. It returns the graph with one vertex for
  each  row of [22XM[122X and an edges between vertices [22Xi[122X and [22Xj[122X if the [22X(i,j)[122X entry in [22XM[122X
  equals 1.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XCayleyGraphOfGroup(G,A)[110X[133X
  
  [33X[0;0YInputs a group [22XG[122X and a set [22XA[122X of generators. It returns the Cayley graph.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XPathComponentsOfGraph(G,n)[110X[133X
  
  [33X[0;0YInputs  a  graph  [22XG[122X  and  a nonnegative integer [22Xn[122X. If [22Xn=0[122X the number of path
  components  is returned. Otherwise the n-th path component is returned (as a
  graph).[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XContractGraph(G)[110X[133X
  
  [33X[0;0YInputs a graph [22XG[122X and tries to remove vertices and edges to produce a smaller
  graph  [22XG'[122X such that the indlusion [22XG' → G[122X induces a homotopy equivalence [22XRG →
  RG'[122X of Rips complexes. If the graph [22XG[122X is modified the function returns true,
  and otherwise returns false.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XGraphDisplay(G)[110X[133X
  
  [33X[0;0YThis function uses GraphViz software to display a graph [22XG[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XSimplicialMap(K,L,f)[110X[133X
  [33X[0;0Y[10XSimplicialMapNC(K,L,f)[110X[133X
  
  [33X[0;0YInputs  simplicial  complexes  [22XK[122X  ,  [22XL[122X  and  a  function  [22Xf:  K!.vertices  →
  L!.vertices[122X  representing  a simplicial map. It returns a simplicial map [22XK →
  L[122X.   If   [22Xf[122X   does   not   happen   to   represent  a  simplicial  map  then
  SimplicialMap(K,L,f)  will  return  fail; SimplicialMapNC(K,L,f) will not do
  any check and always return something of the data type "simplicial map".[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XChainMapOfSimplicialMap(f)[110X[133X
  
  [33X[0;0YInputs  a  simplicial  map  [22Xf: K → L[122X and returns the corresponding chain map
  [22XC_∗(f) : C_∗(K) → C_∗(L)[122X of the simplicial chain complexes..[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XSimplicialNerveOfGraph(G,d)[110X[133X
  
  [33X[0;0YInputs  a  graph  [22XG[122X  and  returns a [22Xd[122X-dimensional simplicial complex [22XK[122X whose
  1-skeleton  is equal to [22XG[122X. There is a simplicial inclusion [22XK → RG[122X where: (i)
  the  inclusion  induces  isomorphisms  on homotopy groups in dimensions less
  than [22Xd[122X; (ii) the complex [22XRG[122X is the Rips complex (with one [22Xn[122X-simplex for each
  complete subgraph of [22XG[122X on [22Xn+1[122X vertices).[133X
  
