  
  [1X11 [33X[0;0YResolutions[133X[101X
  
  [33X[0;0YThere  is  a  range of functions in HAP that input a group [22XG[122X, integer [22Xn[122X, and
  attempt  to  return  the  first  [22Xn[122X  terms of a free [22XZG[122X-resolution [22XR_∗[122X of the
  trivial module [22XZ[122X. In some cases an explicit contracting homotopy is provided
  on the resolution. The function [10XSize(R)[110X returns a list whose [22Xk[122Xth term is the
  sum of the lengths of the boundaries of the generators in degree [22Xk[122X.[133X
  
  
  [1X11.1 [33X[0;0YResolutions for small finite groups[133X[101X
  
  [33X[0;0YThe following uses discrete Morse theory to construct a resolution.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SymmetricGroup(6);; n:=6;;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionFiniteGroup(G,n);[127X[104X
    [4X[28XResolution of length 6 in characteristic 0 for Group([ (1,2), (1,2,3,4,5,6) [128X[104X
    [4X[28X ]) .[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 10, 58, 186, 452, 906, 1436 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.2 [33X[0;0YResolutions for very small finite groups[133X[101X
  
  [33X[0;0YThe following uses linear algebra over [22XZ[122X to construct a resolution.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQ:=QuaternionGroup(128);;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionSmallGroup(Q,20);[127X[104X
    [4X[28XResolution of length 20 in characteristic 0 for <pc group of size 128 with [128X[104X
    [4X[28X2 generators> . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128, 4, 42, 8, 128 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  suspicion  that  this  resolution  [22XR_∗[122X  is  periodic of period [22X4[122X can be
  verified by constructing the chain complex [22XC_∗=R_∗⊗_ Z ZG[122X and verifying that
  boundary matrices repeat with period [22X4[122X.[133X
  
  [33X[0;0YA  second  example of a periodic resolution, for the Dihedral group [22XD_2k+1=⟨
  x,  y  |  x^2= xy^kx^-1y^-k-1⟩[122X of order [22X2k+2[122X in the case [22Xk=1[122X, is constructed
  and verified for periodicity in the next example.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionSmallGroup(D,15);;[127X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 4, 7, 8, 6, 4, 8, 8, 6, 4, 8, 8, 6, 4, 8, 8 ][128X[104X
    [4X[25Xgap>[125X [27XC:=TensorWithIntegersOverSubgroup(R,Group(One(D)));;[127X[104X
    [4X[25Xgap>[125X [27Xn:=4;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xn:=5;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xn:=6;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xn:=7;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xn:=8;;BoundaryMatrix(C,n)=BoundaryMatrix(C,n+4);[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis periodic resolution for [22XD_3[122X can be found in a paper by R. Swan [Swa60].
  The  resolution  was  proved  for arbitrary [22XD_2k+1[122X by Irina Kholodna [Kho01]
  (Corollary  5.5) and is the cellular chain complex of the universal cover of
  a  CW-complex  [22XX[122X  with  two  cells  in dimensions [22X1, 2 mod 4[122X and one cell in
  dimensions  [22X0,3  mod  4[122X.  The  [22X2[122X-skelecton  is  the  [22X2[122X-complex for the given
  presentation of [22XD_2k+1[122X and an attaching map for the [22X3[122X-cell is represented as
  follows.[133X
  
  [33X[0;0YA  slightly  different  periodic  resolution for [22XD_2k+1[122X has been obtain more
  recently   by  FEA  Johnson  [Joh16].  Johnson's  resolution  has  two  free
  generators  in  each  degree.  Interestingly, running the following code for
  many  values  of  [22Xk  >1[122X seems to produce a periodic resolution with two free
  generators in each degree for most values of [22Xk[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xk:=20;;rels:=[x^2,x*y^k*x^-1*y^(-1-k)];;D:=F/rels;;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionSmallGroup(D,7);;[127X[104X
    [4X[25Xgap>[125X [27XList([0..7],R!.dimension);[127X[104X
    [4X[28X[ 1, 2, 2, 2, 2, 2, 2, 2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.3 [33X[0;0YResolutions for finite groups acting on orbit polytopes[133X[101X
  
  [33X[0;0YThe  following  uses  Polymake  convex  hull  computations  and  homological
  perturbation theory to construct a resolution.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SignedPermutationGroup(5);;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription(G);[127X[104X
    [4X[28X"C2 x ((C2 x C2 x C2 x C2) : S5)"[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xv:=[1,2,3,4,5];;  #The resolution depends on the choice of vector.[127X[104X
    [4X[25Xgap>[125X [27XP:=PolytopalComplex(G,[1,2,3,4,5]);[127X[104X
    [4X[28XNon-free resolution in characteristic 0 for <matrix group of size 3840 with [128X[104X
    [4X[28X9 generators> . [128X[104X
    [4X[28XNo contracting homotopy available.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XR:=FreeGResolution(P,6);[127X[104X
    [4X[28XResolution of length 5 in characteristic 0 for <matrix group of size [128X[104X
    [4X[28X3840 with 9 generators> . [128X[104X
    [4X[28XNo contracting homotopy available.[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 10, 60, 214, 694, 6247, 273600 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  convex polytope [22XP_G(v)= Convex~Hull{g⋅ v | g∈ G}[122X used in the resolution
  depends  on  the  choice  of  vector  [22Xv∈  R^n[122X.  Two  such  polytopes for the
  alternating group [22XG=A_4[122X acting on [22XR^4[122X can be visualized as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=AlternatingGroup(4);;[127X[104X
    [4X[25Xgap>[125X [27XOrbitPolytope(G,[1,2,3,4],["VISUAL"]);[127X[104X
    [4X[25Xgap>[125X [27XOrbitPolytope(G,[1,1,3,4],["VISUAL"]);[127X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XP1:=PolytopalComplex(G,[1,2,3,4]);;[127X[104X
    [4X[25Xgap>[125X [27XP2:=PolytopalComplex(G,[1,1,3,4]);;[127X[104X
    [4X[25Xgap>[125X [27XR1:=FreeGResolution(P1,20);;[127X[104X
    [4X[25Xgap>[125X [27XR2:=FreeGResolution(P2,20);;[127X[104X
    [4X[25Xgap>[125X [27XSize(R1);[127X[104X
    [4X[28X[ 6, 11, 32, 24, 36, 60, 65, 102, 116, 168, 172, 248, 323, 628, 650, 1093, [128X[104X
    [4X[28X  1107, 2456, 2344, 6115 ][128X[104X
    [4X[25Xgap>[125X [27XSize(R2);[127X[104X
    [4X[28X[ 4, 11, 20, 24, 36, 60, 65, 102, 116, 168, 172, 248, 323, 628, 650, 1093, [128X[104X
    [4X[28X  1107, 2456, 2344, 6115 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.4 [33X[0;0YMinimal resolutions for finite [22Xp[122X[101X[1X-groups over [22XF_p[122X[101X[1X[133X[101X
  
  [33X[0;0YThe   following   uses   linear   algebra   to   construct  a  minimal  free
  [22XF_pG[122X-resolution of the trivial module [22XF[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XP:=SylowSubgroup(MathieuGroup(12),2);;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionPrimePowerGroup(P,20);[127X[104X
    [4X[28XResolution of length 20 in characteristic 2 for Group([128X[104X
    [4X[28X[ (2,8,4,12)(3,11,7,9), (2,3)(4,7)(6,10)(9,11), (3,7)(6,10)(8,11)(9,12), [128X[104X
    [4X[28X  (1,10)(3,7)(5,6)(8,12), (2,4)(3,7)(8,12)(9,11), (1,5)(6,10)(8,12)(9,11) [128X[104X
    [4X[28X ]) . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 6, 62, 282, 740, 1810, 3518, 6440, 10600, 17040, 24162, 34774, 49874, [128X[104X
    [4X[28X  62416, 81780, 106406, 145368, 172282, 208926, 262938, 320558 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  resolution has the minimum number of generators possible in each degree
  and can be used to guess a formula for the Poincare series[133X
  
  [33X[0;0Y[22XP(x) = Σ_k≥ 0 dim_ F_pH^k(G, F_p)x^k[122X.[133X
  
  [33X[0;0YThe guess is certainly correct for the coefficients of [22Xx^k[122X for [22Xk≤ 20[122X and can
  be used to guess the dimension of say [22XH^2000(G, F_p)[122X.[133X
  
  [33X[0;0YMost likely [22Xdim_ F_2H^2000(G, F_2) = 2001000[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XP:=PoincareSeries(R,20);[127X[104X
    [4X[28X(1)/(-x_1^3+3*x_1^2-3*x_1+1)[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XExpansionOfRationalFunction(P,2000)[2000];[127X[104X
    [4X[28X2001000[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.5 [33X[0;0YResolutions for abelian groups[133X[101X
  
  [33X[0;0YThe  following uses the formula for the tensor product of chain complexes to
  construct a resolution.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:=AbelianPcpGroup([2,4,8,0,0]);;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription(A);[127X[104X
    [4X[28X"Z x Z x C8 x C4 x C2"[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionAbelianGroup(A,10);[127X[104X
    [4X[28XResolution of length 10 in characteristic 0 for Pcp-group with orders [128X[104X
    [4X[28X[ 2, 4, 8, 0, 0 ] . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 14, 90, 296, 680, 1256, 2024, 2984, 4136, 5480, 7016 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.6 [33X[0;0YResolutions for nilpotent groups[133X[101X
  
  [33X[0;0YThe  following  uses  the  NQ package to express the free nilpotent group of
  class  [22X3[122X  on  three  generators  as a Pcp group [22XG[122X, and then uses homological
  perturbation  on  the  lower  central  series to construct a resolution. The
  resolution is used to exhibit [22X2[122X-torsion in [22XH_4(G, Z)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup(3);;[127X[104X
    [4X[25Xgap>[125X [27XG:=Image(NqEpimorphismNilpotentQuotient(F,3));;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionNilpotentGroup(G,5);[127X[104X
    [4X[28XResolution of length 5 in characteristic 0 for Pcp-group with orders [128X[104X
    [4X[28X[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 28, 377, 2377, 9369, 25850 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XHomology(TensorWithIntegers(R),4);[127X[104X
    [4X[28X[ 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  uses homological perturbation on the lower central series to
  construct a resolution for the Sylow [22X2[122X-subgroup [22XP=Syl_2(M_12)[122X of the Mathieu
  simple group [22XM_12[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=MathieuGroup(12);;[127X[104X
    [4X[25Xgap>[125X [27XP:=SylowSubgroup(G,2);;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription(P);[127X[104X
    [4X[28X"((C4 x C4) : C2) : C2"[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionNilpotentGroup(P,9);[127X[104X
    [4X[28XResolution of length 9 in characteristic [128X[104X
    [4X[28X0 for <permutation group with 279 generators> . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 12, 80, 310, 939, 2556, 6768, 19302, 61786, 237068 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.7 [33X[0;0YResolutions for groups with subnormal series[133X[101X
  
  [33X[0;0YThe  following  uses  homological  perturbation  on  a  subnormal  series to
  construct a resolution for the Sylow [22X2[122X-subgroup [22XP=Syl_2(M_12)[122X of the Mathieu
  simple group [22XM_12[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XP:=SylowSubgroup(MathieuGroup(12),2);;[127X[104X
    [4X[25Xgap>[125X [27Xsn:=ElementaryAbelianSeries(P);;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionSubnormalSeries(sn,9);[127X[104X
    [4X[28XResolution of length 9 in characteristic [128X[104X
    [4X[28X0 for <permutation group with 64 generators> . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 12, 78, 288, 812, 1950, 4256, 8837, 18230, 39120 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.8 [33X[0;0YResolutions for groups with normal series[133X[101X
  
  [33X[0;0YThe  following uses homological perturbation on a normal series to construct
  a  resolution  for  the Sylow [22X2[122X-subgroup [22XP=Syl_2(M_12)[122X of the Mathieu simple
  group [22XM_12[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XP:=SylowSubgroup(MathieuGroup(12),2);;[127X[104X
    [4X[25Xgap>[125X [27XP1:=EfficientNormalSubgroups(P)[1];;[127X[104X
    [4X[25Xgap>[125X [27XP2:=Intersection(DerivedSubgroup(P),P1);;[127X[104X
    [4X[25Xgap>[125X [27XP3:=Group(One(P));;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionNormalSeries([P,P1,P2,P3],9);[127X[104X
    [4X[28XResolution of length 9 in characteristic [128X[104X
    [4X[28X0 for <permutation group with 64 generators> . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 10, 60, 200, 532, 1238, 2804, 6338, 15528, 40649 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.9 [33X[0;0YResolutions for polycyclic (almost) crystallographic groups[133X[101X
  
  [33X[0;0YThe  following  uses  the Polycyclic package and homological perturbation to
  construct a resolution for the crystallographic group [10XG:=SpaceGroup(3,165)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SpaceGroup(3,165);;[127X[104X
    [4X[25Xgap>[125X [27XG:=Image(IsomorphismPcpGroup(G));;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionAlmostCrystalGroup(G,20);[127X[104X
    [4X[28XResolution of length 20 in characteristic 0 for Pcp-group with orders [128X[104X
    [4X[28X[ 3, 2, 0, 0, 0 ] . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 10, 49, 117, 195, 273, 351, 429, 507, 585, 663, 741, 819, 897, 975, 1053, [128X[104X
    [4X[28X  1131, 1209, 1287, 1365, 1443 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  constructs  a  resolution for an almost crystallographic Pcp
  group  [22XG[122X.  The  final  commands  establish  that  [22XG[122X  is  not isomorphic to a
  crystallographic group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=AlmostCrystallographicPcpGroup( 4, 50, [ 1, -4, 1, 2 ] );;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionAlmostCrystalGroup(G,20);[127X[104X
    [4X[28XResolution of length 20 in characteristic 0 for Pcp-group with orders [128X[104X
    [4X[28X[ 4, 0, 0, 0, 0 ] . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 10, 53, 137, 207, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, 223, [128X[104X
    [4X[28X  223, 223, 223, 223, 223 ][128X[104X
    [4X[28X[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XT:=Kernel(NaturalHomomorphismOnHolonomyGroup(G));;[127X[104X
    [4X[25Xgap>[125X [27XIsAbelian(T);[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.10 [33X[0;0YResolutions for Bieberbach groups[133X[101X
  
  [33X[0;0YThe   following   constructs   a   resolution   for   the  Bieberbach  group
  [10XG=SpaceGroup(3,165)[110X by using convex hull algorithms to construct a Dirichlet
  domain  for  its  free  action  on  Euclidean space [22XR^3[122X. By construction the
  resolution is trivial in degrees [22X≥ 3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SpaceGroup(3,165);;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionBieberbachGroup(G);[127X[104X
    [4X[28XResolution of length 4 in characteristic [128X[104X
    [4X[28X0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 10, 18, 8, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe   fundamental  domain  constructed  for  the  above  resolution  can  be
  visualized using the following commands.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FundamentalDomainBieberbachGroup(G);[127X[104X
    [4X[28X<polymake object>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(F);[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YA  different  fundamental  domain  and  resolution  for [22XG[122X can be obtained by
  changing  the  choice  of  vector  [22Xv∈ R^3[122X in the definition of the Dirichlet
  domain[133X
  
  [33X[0;0Y[22XD(v) = {x∈ R^3 | ||x-v|| ≤ ||x-g.v|| for~all~ g∈ G}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionBieberbachGroup(G,[1/2,1/2,1/2]);[127X[104X
    [4X[28XResolution of length 4 in characteristic [128X[104X
    [4X[28X0 for SpaceGroupOnRightBBNWZ( 3, 6, 1, 1, 4 ) . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 28, 42, 16, 0 ][128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XF:=FundamentalDomainBieberbachGroup(G);[127X[104X
    [4X[28X<polymake object>[128X[104X
    [4X[25Xgap>[125X [27XDisplay(F);[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YA  higher  dimensional example is handled in the next session. A list of the
  [22X62[122X  [22X7[122X-dimensional  Hantze-Wendt Bieberbach groups is loaded and a resolution
  is computed for the first group in the list.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfile:=HapFile("HW-7dim.txt");;[127X[104X
    [4X[25Xgap>[125X [27XRead(file);[127X[104X
    [4X[25Xgap>[125X [27XG:=HWO7Gr[1];[127X[104X
    [4X[28X<matrix group with 7 generators>[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionBieberbachGroup(G);[127X[104X
    [4X[28XResolution of length 8 in characteristic 0 for <matrix group with [128X[104X
    [4X[28X7 generators> . [128X[104X
    [4X[28XNo contracting homotopy available.[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 284, 1512, 3780, 4480, 2520, 840, 84, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  homological  perturbation  techniques  needed  to extend this method to
  crystallographic   groups   acting  non-freely  on  [22XR^n[122X  has  not  yet  been
  implemenyed. This is on the TO-DO list.[133X
  
  
  [1X11.11 [33X[0;0YResolutions for arbitrary crystallographic groups[133X[101X
  
  [33X[0;0YAn  implementation  of  the  above  method  for  Bieberbach  groups  is also
  available  for  arbitrary  crystallographic  groups.  The  following example
  constructs a resolution for the group [10XG:=SpaceGroupIT(3,227)[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SpaceGroupIT(3,227);;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionSpaceGroup(G,11);[127X[104X
    [4X[28XResolution of length 11 in characteristic 0 for <matrix group with [128X[104X
    [4X[28X8 generators> . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 38, 246, 456, 644, 980, 1427, 2141, 2957, 3993, 4911, 6179 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.12  [33X[0;0YResolutions for crystallographic groups admitting cubical fundamental[101X
  [1Xdomain[133X[101X
  
  [33X[0;0YThe  following uses subdivision techniques to construct a resolution for the
  Bieberbach  group  [10XG:=SpaceGroup(4,122)[110X.  The  resolution  is endowed with a
  contracting homotopy.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SpaceGroup(4,122);;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionCubicalCrystGroup(G,20);[127X[104X
    [4X[28XResolution of length 20 in characteristic 0 for <matrix group with [128X[104X
    [4X[28X6 generators> . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 8, 24, 24, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSubdivision and homological perturbation are used to construct the following
  resolution  (with  contracting  homotopy)  for a crystallographic group with
  non-free action.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=SpaceGroup(4,1100);;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionCubicalCrystGroup(G,20);[127X[104X
    [4X[28XResolution of length 20 in characteristic 0 for <matrix group with [128X[104X
    [4X[28X8 generators> . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 40, 215, 522, 738, 962, 1198, 1466, 1734, 2034, 2334, 2666, 2998, 3362, [128X[104X
    [4X[28X  3726, 4122, 4518, 4946, 5374, 5834, 6294 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.13 [33X[0;0YResolutions for Coxeter groups[133X[101X
  
  [33X[0;0YThe  following  session constructs the Coxeter diagram for the Coxeter group
  [22XB=B_7[122X of order [22X645120[122X. A resolution for [22XG[122X is then computed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XD:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,3]],[5,[6,3]],[6,[7,4]]];;[127X[104X
    [4X[25Xgap>[125X [27XCoxeterDiagramDisplay(D);;[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionCoxeterGroup(D,5);[127X[104X
    [4X[28XResolution of length 5 in characteristic [128X[104X
    [4X[28X0 for <permutation group of size 645120 with 7 generators> . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 14, 112, 492, 1604, 5048 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  routine  extension  of this method to infinite Coxeter groups is on the
  TO-DO list.[133X
  
  
  [1X11.14 [33X[0;0YResolutions for Artin groups[133X[101X
  
  [33X[0;0YThe following session constructs a resolution for the infinite Artin group [22XG[122X
  associated  to the Coxeter group [22XB_7[122X. Exactness of the resolution depends on
  the solution to the [22XK(π,1)[122X Conjecture for Artin groups of spherical type.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionArtinGroup(D,8);[127X[104X
    [4X[28XResolution of length 8 in characteristic 0 for <fp group on the generators [128X[104X
    [4X[28X[ f1, f2, f3, f4, f5, f6, f7 ]> . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 14, 98, 310, 610, 918, 1326, 2186, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.15 [33X[0;0YResolutions for [22XG=SL_2( Z[1/m])[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  following  uses  homological perturbation to construct a resolution for
  [22XG=SL_2( Z[1/6])[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionSL2Z(6,10);[127X[104X
    [4X[28XResolution of length 10 in characteristic 0 for SL(2,Z[1/6]) . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 44, 679, 6910, 21304, 24362, 48506, 43846, 90928, 86039, 196210 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.16 [33X[0;0YResolutions for selected groups [22XG=SL_2( mathcal O( Q(sqrtd) )[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  following  uses finite "Voronoi complexes" and homological perturbation
  to  construct  a  resolution  for  [22XG=SL_2(mathcal  O( Q(sqrt-5))[122X. The finite
  complexes  were contributed independently by A. Rahm, M. Dutour-Scikiric and
  S.      Schoenenbeck      and      are      stored     in     the     folder
  [10X~pkg/Hap1.v/lib/Perturbations/Gcomplexes[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionSL2QuadraticIntegers(-5,10);[127X[104X
    [4X[28XResolution of length 10 in characteristic 0 for matrix group . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 22, 114, 120, 200, 146, 156, 136, 254, 168, 170 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.17 [33X[0;0YResolutions for selected groups [22XG=PSL_2( mathcal O( Q(sqrtd) )[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  following  uses finite "Voronoi complexes" and homological perturbation
  to  construct  a  resolution  for [22XG=PSL_2(mathcal O( Q(sqrt-11))[122X. The finite
  complexes  were contributed independently by A. Rahm, M. Dutour-Scikiric and
  S.      Schoenenbeck      and      are      stored     in     the     folder
  [10X~pkg/Hap1.v/lib/Perturbations/Gcomplexes[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionPSL2QuadraticIntegers(-11,10);[127X[104X
    [4X[28XResolution of length 10 in characteristic 0 for PSL(2,O-11) . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 12, 59, 89, 107, 125, 230, 208, 270, 326, 515 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.18 [33X[0;0YResolutions for a few higher-dimensional arithmetic groups[133X[101X
  
  [33X[0;0YThe  following  uses finite "Voronoi complexes" and homological perturbation
  to  construct  a  resolution  for  [22XG=PSL_4(  Z)[122X.  The  finite complexes were
  contributed   by   M.   Dutour-Scikiric   and   are  stored  in  the  folder
  [10X~pkg/Hap1.v/lib/Perturbations/Gcomplexes[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X V:=ContractibleGcomplex("PSL(4,Z)_d");[127X[104X
    [4X[28XNon-free resolution in characteristic 0 for matrix group . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XR:=FreeGResolution(V,5);[127X[104X
    [4X[28XResolution of length 5 in characteristic 0 for matrix group . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 18, 210, 1444, 26813 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.19 [33X[0;0YResolutions for finite-index subgroups[133X[101X
  
  [33X[0;0YThe  next  commands  first construct the congruence subgroup [22XΓ_0(I)[122X of index
  [22X144[122X in [22XSL_2(cal O Q(sqrt-2))[122X for the ideal [22XI[122X in [22Xcal O Q(sqrt-2)[122X generated by
  [22X4+5sqrt-2[122X.  The  commands  then  compute  a  resolution  for  the congruence
  subgroup [22XG=Γ_0(I) ≤ SL_2(cal O Q(sqrt-2))[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQ:=QuadraticNumberField(-2);;[127X[104X
    [4X[25Xgap>[125X [27XOQ:=RingOfIntegers(Q);;[127X[104X
    [4X[25Xgap>[125X [27XI:=QuadraticIdeal(OQ,4+5*Sqrt(-2));;[127X[104X
    [4X[25Xgap>[125X [27XG:=HAP_CongruenceSubgroupGamma0(I);[127X[104X
    [4X[28X<[group of 2x2 matrices in characteristic 0>[128X[104X
    [4X[25Xgap>[125X [27X[127X[104X
    [4X[25Xgap>[125X [27XIndexInSL2O(G);[127X[104X
    [4X[28X144[128X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionSL2QuadraticIntegers(-2,4,true);;[127X[104X
    [4X[25Xgap>[125X [27XS:=ResolutionFiniteSubgroup(R,G);[127X[104X
    [4X[28XResolution of length 4 in characteristic 0 for <matrix group with [128X[104X
    [4X[28X290 generators> . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X[ 1152, 8496, 30960, 59616 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.20 [33X[0;0YSimplifying resolutions[133X[101X
  
  [33X[0;0YThe next commands construct a resolution [22XR_∗[122X for the symmetric group [22XS_5[122X and
  convert  it  to a resolution [22XS_∗[122X for the finite index subgroup [22XA_4 < S_5[122X. An
  heuristic  algorithm  is  applied  to [22XS_∗[122X in the hope of obtaining a smaller
  resolution [22XT_∗[122X for the alternating group [22XA_4[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionFiniteGroup(SymmetricGroup(5),5);;[127X[104X
    [4X[25Xgap>[125X [27XS:=ResolutionFiniteSubgroup(R,AlternatingGroup(4));[127X[104X
    [4X[28XResolution of length 5 in characteristic 0 for Alt( [ 1 .. 4 ] ) . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(S);[127X[104X
    [4X[28X[ 80, 380, 1000, 2040, 3400 ][128X[104X
    [4X[25Xgap>[125X [27XT:=SimplifiedComplex(S);[127X[104X
    [4X[28XResolution of length 5 in characteristic 0 for Alt( [ 1 .. 4 ] ) . [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(T);[127X[104X
    [4X[28X[ 4, 34, 22, 19, 196 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.21  [33X[0;0YResolutions  for  graphs  of  groups  and  for groups with aspherical[101X
  [1Xpresentations[133X[101X
  
  [33X[0;0YThe following example constructs a resolution for a finitely presented group
  whose  presentation  is  known  to  have  the  property  that its associated
  [22X2[122X-complex is aspherical.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup(3);;x:=F.1;;y:=F.2;;z:=F.3;;[127X[104X
    [4X[25Xgap>[125X [27Xrels:=[x*y*x*(y*x*y)^-1, y*z*y*(z*y*z)^-1, z*x*z*(x*z*x)^-1];;[127X[104X
    [4X[25Xgap>[125X [27XG:=F/rels;;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionAsphericalPresentation(G,10);[127X[104X
    [4X[28XResolution of length 10 in characteristic 0 for <fp group on the generators [128X[104X
    [4X[28X[ f1, f2, f3 ]> . [128X[104X
    [4X[28XNo contracting homotopy available. [128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 6, 18, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe   following   commands  create  a  resolution  for  a  graph  of  groups
  corresponding  to  the  amalgamated  product  [22XG=H∗_AK[122X  where  [22XH=S_5[122X  is  the
  symmetric  group  of  degree [22X5[122X, [22XK=S_4[122X is the symmetric group of degree [22X4[122X and
  the common subgroup is [22XA=S_3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS5:=SymmetricGroup(5);SetName(S5,"S5");;[127X[104X
    [4X[28XSym( [ 1 .. 5 ] )[128X[104X
    [4X[25Xgap>[125X [27XS4:=SymmetricGroup(4);SetName(S4,"S4");;[127X[104X
    [4X[28XSym( [ 1 .. 4 ] )[128X[104X
    [4X[25Xgap>[125X [27XA:=SymmetricGroup(3);SetName(A,"S3");;[127X[104X
    [4X[28XSym( [ 1 .. 3 ] )[128X[104X
    [4X[25Xgap>[125X [27XAS5:=GroupHomomorphismByFunction(A,S5,x->x);;[127X[104X
    [4X[25Xgap>[125X [27XAS4:=GroupHomomorphismByFunction(A,S4,x->x);;[127X[104X
    [4X[25Xgap>[125X [27XD:=[S5,S4,[AS5,AS4]];;[127X[104X
    [4X[25Xgap>[125X [27XGraphOfGroupsDisplay(D);;[127X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionGraphOfGroups(D,8);;[127X[104X
    [4X[25Xgap>[125X [27XSize(R);[127X[104X
    [4X[28X[ 16, 68, 162, 302, 480, 627, 869, 1290 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X11.22 [33X[0;0YResolutions for [22XFG[122X[101X[1X-modules[133X[101X
  
  [33X[0;0YLet  [22XF=  F_p[122X  be the field of [22Xp[122X elements and let [22XM[122X be some [22XFG[122X-module for [22XG[122X a
  finite  [22Xp[122X-group.  We  might wish to construct a free [22XFG[122X-resolution for [22XM[122X. We
  can handle this by constructing a short exact sequence[133X
  
  [33X[0;0Y[22XDM ↣ P ↠ M[122X[133X
  
  [33X[0;0Yin  which  [22XP[122X  is  free  (or  projective). Then any resolution of [22XDM[122X yields a
  resolution  of [22XM[122X and we can represent [22XDM[122X as a submodule of [22XP[122X. We refer to [22XDM[122X
  as the [13Xdesuspension[113X of [22XM[122X. Consider for instance [22XG=Syl_2(GL(4,2))[122X and [22XF= F_2[122X.
  The  matrix  group [22XG[122X acts via matrix multiplication on [22XM= F^4[122X. The following
  example constructs a free [22XFG[122X-resolution for [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=GL(4,2);;[127X[104X
    [4X[25Xgap>[125X [27XS:=SylowSubgroup(G,2);;[127X[104X
    [4X[25Xgap>[125X [27XM:=GModuleByMats(GeneratorsOfGroup(S),GF(2));;[127X[104X
    [4X[25Xgap>[125X [27XDM:=DesuspensionMtxModule(M);;[127X[104X
    [4X[25Xgap>[125X [27XR:=ResolutionFpGModule(DM,20);[127X[104X
    [4X[28XResolution of length 20 in characteristic 2 for <matrix group of [128X[104X
    [4X[28Xsize 64 with 3 generators> .[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XList([0..20],R!.dimension);[127X[104X
    [4X[28X[ 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, [128X[104X
    [4X[28X153, 171, 190, 210, 231, 253 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
