  
  [1X21 [33X[0;0YSimplicial groups[133X[101X
  
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XNerveOfCatOneGroup(G,n)[110X[133X
  
  [33X[0;0YInputs   a   cat-1-group  [22XG[122X  and  a  positive  integer  [22Xn[122X.  It  returns  the
  low-dimensional part of the nerve of [22XG[122X as a simplicial group of length [22Xn[122X.[133X
  [33X[0;0YThis  function applies both to cat-1-groups for which IsHapCatOneGroup(G) is
  true, and to cat-1-groups produced using the Xmod package.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XEilenbergMacLaneSimplicialGroup(G,n,dim)[110X[133X
  
  [33X[0;0YInputs  a  group  [22XG[122X,  a  positive integer [22Xn[122X, and a positive integer [22Xdim[122X. The
  function  returns  the  first  [22X1+dim[122X  terms of a simplicial group with [22Xn-1[122Xst
  homotopy group equal to [22XG[122X and all other homotopy groups equal to zero.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XEilenbergMacLaneSimplicialGroupMap(f,n,dim)[110X[133X
  
  [33X[0;0YInputs  a  group  homomorphism  [22Xf:G→ Q[122X, a positive integer [22Xn[122X, and a positive
  integer  [22Xdim[122X.  The  function  returns  the first [22X1+dim[122X terms of a simplicial
  group homomorphism [22Xf:K(G,n) → K(Q,n)[122X of Eilenberg-MacLane simplicial groups.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XMooreComplex(G)[110X[133X
  
  [33X[0;0YInputs a simplicial group [22XG[122X and returns its Moore complex as a [22XG[122X-complex.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XChainComplexOfSimplicialGroup(G)[110X[133X
  
  [33X[0;0YInputs  a  simplicial  group [22XG[122X and returns the cellular chain complex [22XC[122X of a
  CW-space  [22XX[122X  represented  by the homotopy type of the simplicial group. Thus
  the homology groups of [22XC[122X are the integral homology groups of [22XX[122X.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XSimplicialGroupMap(f)[110X[133X
  
  [33X[0;0YInputs  a  homomorphism [22Xf:G→ Q[122X of simplicial groups. The function returns an
  induced  map [22Xf:C(G) → C(Q)[122X of chain complexes whose homology is the integral
  homology of the simplicial group G and Q respectively.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XHomotopyGroup(G,n)[110X[133X
  
  [33X[0;0YInputs  a simplicial group [22XG[122X and a positive integer [22Xn[122X. The integer [22Xn[122X must be
  less than the length of [22XG[122X. It returns, as a group, the (n)-th homology group
  of  its  Moore  complex.  Thus  HomotopyGroup(G,0)  returns the "fundamental
  group" of [22XG[122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XRepresentation of elements in the bar resolution[110X[133X
  
  [33X[0;0YFor  a  group  G we denote by [22XB_n(G)[122X the free [22XZG[122X-module with basis the lists
  [22X[g_1 | g_2 | ... | g_n][122X where the [22Xg_i[122X range over [22XG[122X.[133X
  [33X[0;0YWe represent a word[133X
  [33X[0;0Y[22Xw  = h_1.[g_11 | g_12 | ... | g_1n] - h_2.[g_21 | g_22 | ... | g_2n] + ... +
  h_k.[g_k1 | g_k2 | ... | g_kn][122X[133X
  [33X[0;0Yin [22XB_n(G)[122X as a list of lists:[133X
  [33X[0;0Y[22X[  [+1,h_1,g_11  , g_12 , ... , g_1n] , [-1, h_2,g_21 , g_22 , ... | g_2n] +
  ... + [+1, h_k,g_k1 , g_k2 , ... , g_kn][122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XBarResolutionBoundary(w)[110X[133X
  
  [33X[0;0YThis  function  inputs  a  word  [22Xw[122X  in  the bar resolution module [22XB_n(G)[122X and
  returns  its image under the boundary homomorphism [22Xd_n: B_n(G) → B_n-1(G)[122X in
  the bar resolution.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XBarResolutionHomotopy(w)[110X[133X
  
  [33X[0;0YThis  function  inputs  a  word  [22Xw[122X  in  the bar resolution module [22XB_n(G)[122X and
  returns  its  image under the contracting homotopy [22Xh_n: B_n(G) → B_n+1(G)[122X in
  the bar resolution.[133X
  [33X[0;0YThis function is currently being implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XRepresentation of elements in the bar complex[110X[133X
  
  [33X[0;0YFor  a  group  G  we denote by [22XBC_n(G)[122X the free abelian group with basis the
  lists [22X[g_1 | g_2 | ... | g_n][122X where the [22Xg_i[122X range over [22XG[122X.[133X
  [33X[0;0YWe represent a word[133X
  [33X[0;0Y[22Xw  = [g_11 | g_12 | ... | g_1n] - [g_21 | g_22 | ... | g_2n] + ... + [g_k1 |
  g_k2 | ... | g_kn][122X[133X
  [33X[0;0Yin [22XBC_n(G)[122X as a list of lists:[133X
  [33X[0;0Y[22X[  [+1,g_11  ,  g_12  , ... , g_1n] , [-1, g_21 , g_22 , ... | g_2n] + ... +
  [+1, g_k1 , g_k2 , ... , g_kn][122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XBarComplexBoundary(w)[110X[133X
  
  [33X[0;0YThis  function  inputs  a word [22Xw[122X in the n-th term of the bar complex [22XBC_n(G)[122X
  and  returns  its  image  under  the  boundary  homomorphism  [22Xd_n: BC_n(G) →
  BC_n-1(G)[122X in the bar complex.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XBarResolutionEquivalence(R)[110X[133X
  
  [33X[0;0YThis  function  inputs a free [22XZG[122X-resolution [22XR[122X. It returns a component object
  HE with components[133X
  [33X[0;0YHE!.phi(n,w)  is a function which inputs a non-negative integer [22Xn[122X and a word
  [22Xw[122X  in  [22XB_n(G)[122X. It returns the image of [22Xw[122X in [22XR_n[122X under a chain equivalence [22Xϕ:
  B_n(G) → R_n[122X.[133X
  [33X[0;0YHE!.psi(n,w)  is a function which inputs a non-negative integer [22Xn[122X and a word
  [22Xw[122X  in  [22XR_n[122X. It returns the image of [22Xw[122X in [22XB_n(G)[122X under a chain equivalence [22Xψ:
  R_n → B_n(G)[122X.[133X
  [33X[0;0YHE!.equiv(n,w)  is  a  function  which inputs a non-negative integer [22Xn[122X and a
  word  [22Xw[122X  in  [22XB_n(G)[122X.  It  returns  the  image  of  [22Xw[122X  in  [22XB_n+1(G)[122X  under  a
  [22XZG[122X-equivariant homomorphism[133X
  [33X[0;0Y[22Xequiv(n,-) : B_n(G) → B_n+1(G)[122X[133X
  [33X[0;0Ysatisfying[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yw - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) .[133X [124X[133X
  
  
  [33X[0;0Ywhere  [22Xd(n,-):  B_n(G)  →  B_n-1(G)[122X  is the boundary homomorphism in the bar
  resolution.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XBarComplexEquivalence(R)[110X[133X
  
  [33X[0;0YThis  function  inputs a free [22XZG[122X-resolution [22XR[122X. It first constructs the chain
  complex [22XT=TensorWithIntegerts(R)[122X. The function returns a component object HE
  with components[133X
  [33X[0;0YHE!.phi(n,w)  is a function which inputs a non-negative integer [22Xn[122X and a word
  [22Xw[122X  in [22XBC_n(G)[122X. It returns the image of [22Xw[122X in [22XT_n[122X under a chain equivalence [22Xϕ:
  BC_n(G) → T_n[122X.[133X
  [33X[0;0YHE!.psi(n,w)  is  a  function  which  inputs a non-negative integer [22Xn[122X and an
  element  [22Xw[122X  in  [22XT_n[122X.  It  returns  the  image  of [22Xw[122X in [22XBC_n(G)[122X under a chain
  equivalence [22Xψ: T_n → BC_n(G)[122X.[133X
  [33X[0;0YHE!.equiv(n,w)  is  a  function  which inputs a non-negative integer [22Xn[122X and a
  word  [22Xw[122X  in  [22XBC_n(G)[122X.  It  returns  the  image  of  [22Xw[122X  in  [22XBC_n+1(G)[122X under a
  homomorphism[133X
  [33X[0;0Y[22Xequiv(n,-) : BC_n(G) → BC_n+1(G)[122X[133X
  [33X[0;0Ysatisfying[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yw - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) .[133X [124X[133X
  
  
  [33X[0;0Ywhere  [22Xd(n,-):  BC_n(G)  → BC_n-1(G)[122X is the boundary homomorphism in the bar
  complex.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XRepresentation of elements in the bar cocomplex[110X[133X
  
  [33X[0;0YFor  a  group  G  we denote by [22XBC^n(G)[122X the free abelian group with basis the
  lists [22X[g_1 | g_2 | ... | g_n][122X where the [22Xg_i[122X range over [22XG[122X.[133X
  [33X[0;0YWe represent a word[133X
  [33X[0;0Y[22Xw  = [g_11 | g_12 | ... | g_1n] - [g_21 | g_22 | ... | g_2n] + ... + [g_k1 |
  g_k2 | ... | g_kn][122X[133X
  [33X[0;0Yin [22XBC^n(G)[122X as a list of lists:[133X
  [33X[0;0Y[22X[  [+1,g_11  ,  g_12  , ... , g_1n] , [-1, g_21 , g_22 , ... | g_2n] + ... +
  [+1, g_k1 , g_k2 , ... , g_kn][122X.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XBarCocomplexCoboundary(w)[110X[133X
  
  [33X[0;0YThis  function inputs a word [22Xw[122X in the n-th term of the bar cocomplex [22XBC^n(G)[122X
  and  returns  its  image  under  the  coboundary homomorphism [22Xd^n: BC^n(G) →
  BC^n+1(G)[122X in the bar cocomplex.[133X
  [33X[0;0YThis function was implemented by [12XVan Luyen Le[112X.[133X
  
