  
  [1X25 [33X[0;0YSimplicial groups[133X[101X
  
  
  [1X25.1 [33X[0;0Y [133X[101X
  
  [1X25.1-1 NerveOfCatOneGroup[101X
  
  [33X[1;0Y[29X[2XNerveOfCatOneGroup[102X( [3XG[103X, [3Xn[103X ) [32X function[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[12XExamples:[112X             1             ([7X../tutorial/chap9.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutSimplicialGroups.html[107X) ,                      3
  ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X25.1-2 EilenbergMacLaneSimplicialGroup[101X
  
  [33X[1;0Y[29X[2XEilenbergMacLaneSimplicialGroup[102X( [3XG[103X, [3Xn[103X, [3Xdim[103X ) [32X function[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[12XExamples:[112X   1   ([7X../tutorial/chap3.html[107X) ,  2  ([7X../tutorial/chap9.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutCoveringSpaces.html[107X) ,                        4
  ([7X../www/SideLinks/About/aboutCoverinSpaces.html[107X) ,                         5
  ([7X../www/SideLinks/About/aboutSimplicialGroups.html[107X) [133X
  
  [1X25.1-3 EilenbergMacLaneSimplicialGroupMap[101X
  
  [33X[1;0Y[29X[2XEilenbergMacLaneSimplicialGroupMap[102X [32X global variable[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[12XExamples:[112X[133X
  
  [1X25.1-4 MooreComplex[101X
  
  [33X[1;0Y[29X[2XMooreComplex[102X( [3XG[103X ) [32X function[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[12XExamples:[112X[133X
  
  [1X25.1-5 ChainComplexOfSimplicialGroup[101X
  
  [33X[1;0Y[29X[2XChainComplexOfSimplicialGroup[102X( [3XG[103X ) [32X function[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[12XExamples:[112X   1   ([7X../tutorial/chap3.html[107X) ,  2  ([7X../tutorial/chap9.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutCoveringSpaces.html[107X) ,                        4
  ([7X../www/SideLinks/About/aboutCoverinSpaces.html[107X) ,                         5
  ([7X../www/SideLinks/About/aboutSimplicialGroups.html[107X) ,                      6
  ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X25.1-6 SimplicialGroupMap[101X
  
  [33X[1;0Y[29X[2XSimplicialGroupMap[102X [32X global variable[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[12XExamples:[112X[133X
  
  [1X25.1-7 HomotopyGroup[101X
  
  [33X[1;0Y[29X[2XHomotopyGroup[102X( [3XG[103X, [3Xn[103X ) [32X function[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[12XExamples:[112X   1   ([7X../tutorial/chap5.html[107X) ,  2  ([7X../tutorial/chap9.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutNonabelian.html[107X) ,                            4
  ([7X../www/SideLinks/About/aboutCrossedMods.html[107X) ,                           5
  ([7X../www/SideLinks/About/aboutquasi.html[107X) ,                                 6
  ([7X../www/SideLinks/About/aboutSimplicialGroups.html[107X) ,                      7
  ([7X../www/SideLinks/About/aboutIntro.html[107X) ,                                 8
  ([7X../www/SideLinks/About/aboutTensorSquare.html[107X) [133X
  
  [1X25.1-8 Representation of elements in the bar resolution[101X
  
  [33X[1;0Y[29X[2XRepresentation of elements in the bar resolution[102X [32X global variable[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[12XExamples:[112X[133X
  
  [1X25.1-9 BarResolutionBoundary[101X
  
  [33X[1;0Y[29X[2XBarResolutionBoundary[102X [32X global variable[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[12XExamples:[112X[133X
  
  [1X25.1-10 BarResolutionHomotopy[101X
  
  [33X[1;0Y[29X[2XBarResolutionHomotopy[102X [32X global variable[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[12XExamples:[112X[133X
  
  [1X25.1-11 Representation of elements in the bar complex[101X
  
  [33X[1;0Y[29X[2XRepresentation of elements in the bar complex[102X [32X global variable[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[12XExamples:[112X[133X
  
  [1X25.1-12 BarComplexBoundary[101X
  
  [33X[1;0Y[29X[2XBarComplexBoundary[102X [32X global variable[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[12XExamples:[112X[133X
  
  [1X25.1-13 BarResolutionEquivalence[101X
  
  [33X[1;0Y[29X[2XBarResolutionEquivalence[102X( [3XR[103X ) [32X function[133X
  
  [33X[0;0YThis  function  inputs a free [22XZG[122X-resolution [22XR[122X. It returns a component object
  HE with components[133X
  
  [30X    [33X[0;6YHE!.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
  
  [30X    [33X[0;6YHE!.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
  
  [30X    [33X[0;6YHE!.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;6Y[22Xequiv(n,-) : B_n(G) → B_n+1(G)[122X[133X
        [33X[0;6Ysatisfying[133X
  
  
  [24X      [33X[0;6Yw - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) .[133X
  
  [124X
  
        [33X[0;6Ywhere  [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[12XExamples:[112X[133X
  
  [1X25.1-14 BarComplexEquivalence[101X
  
  [33X[1;0Y[29X[2XBarComplexEquivalence[102X( [3XR[103X ) [32X function[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
  
  [30X    [33X[0;6YHE!.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
  
  [30X    [33X[0;6YHE!.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
  
  [30X    [33X[0;6YHE!.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;6Y[22Xequiv(n,-) : BC_n(G) → BC_n+1(G)[122X[133X
        [33X[0;6Ysatisfying[133X
  
  
  [24X      [33X[0;6Yw - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) .[133X
  
  [124X
  
        [33X[0;6Ywhere  [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[12XExamples:[112X[133X
  
  [1X25.1-15 Representation of elements in the bar cocomplex[101X
  
  [33X[1;0Y[29X[2XRepresentation of elements in the bar cocomplex[102X [32X global variable[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[12XExamples:[112X[133X
  
  [1X25.1-16 BarCocomplexCoboundary[101X
  
  [33X[1;0Y[29X[2XBarCocomplexCoboundary[102X [32X global variable[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
  
  [33X[0;0Y[12XExamples:[112X[133X
  
