  
  [1X3 [33X[0;0YHomological Group Theory[133X[101X
  
  [33X[0;0YCocycles[133X
  
      │ CcGroup(N,f):: GOuterGroup, StandardCocycle --> CcGroup [33X[0;6YInputs a [22XG[122X-outer group [22XN[122X with nonabelian cocycle describing some extension [22XN ↣ E ↠ G[122X together with standard 2-cocycle [22Xf: G × G → A[122X where [22XA=Z(N)[122X. It returns the extension group determined by the cocycle [22Xf[122X. The group is returned as a cocyclic group.[133X [33X[0;6YThis function is part of the HAPcocyclic package of functions implemented by Robert F. Morse.[133X                                                                                                                 │ 
      │ CocycleCondition(R,n):: FreeRes, Int --> IntMat [33X[0;6YInputs a free [22XZG[122X-resolution [22XR[122X of [22XZ[122X and an integer [22Xn ge 1[122X. It returns an integer matrix [22XM[122X with the following property. Let [22Xd[122X be the [22XZG[122X-rank of [22XR_n[122X. An integer vector [22Xf=[f_1, ... , f_d][122X then represents a [22XZG[122X-homomorphism [22XR_n → Z_q[122X which sends the [22Xi[122Xth generator of [22XR_n[122X to the integer [22Xf_i[122X in the trivial [22XZG[122X-module [22XZ_q= Z/q Z[122X (where possibly [22Xq=0[122X). The homomorphism [22Xf[122X is a cocycle if and only if [22XM^tf=0[122X mod [22Xq[122X.[133X                                                            │ 
      │ StandardCocycle(R,f,n):: FreeRes, List, Int --> Function StandardCocycle(R,f,n,q):: FreeRes, List, Int --> Function [33X[0;6YInputs a free [22XZG[122X-resolution [22XR[122X (with contracting homotopy), a positive integer [22Xn[122X and an integer vector [22Xf[122X representing an [22Xn[122X-cocycle [22XR_n → Z_q= Z/q Z[122X where [22XG[122X acts trivially on [22XZ_q[122X. It is assumed [22Xq=0[122X unless a value for [22Xq[122X is entered. The command returns a function [22XF(g_1, ..., g_n)[122X which is the standard cocycle [22XG^n → Z_q[122X corresponding to [22Xf[122X. At present the command is implemented only for [22Xn=2[122X or [22X3[122X.[133X │ 
  
  [33X[0;0YG-Outer Groups[133X
  
      │ ActedGroup(M):: GOuterGroup --> Group [33X[0;6YInputs a [22XG[122X-outer group [22XM[122X corresponding to a homomorphism [22Xα: G→ Out(N)[122X and returns the group $N$.[133X                                                                                                                                                                                                                                                                                                                                        │ 
      │ ActingGroup(M):: GOuterGroup --> Group [33X[0;6YInputs a [22XG[122X-outer group [22XM[122X corresponding to a homomorphism [22Xα: G→ Out(N)[122X and returns the group $G$.[133X                                                                                                                                                                                                                                                                                                                                       │ 
      │ Centre(M):: GOuterGroup --> GOuterGroup [33X[0;6YInputs a [22XG[122X-outer group [22XM[122X and returns its group-theoretic centre as a [22XG[122X-outer group.[133X                                                                                                                                                                                                                                                                                                                                                   │ 
      │ GOuterGroup(E,N):: Group, Subgroup --> GOuterGroup GOuterGroup():: Group, Subgroup --> GOuterGroup [33X[0;6YInputs a group [22XE[122X and normal subgroup [22XN[122X. It returns [22XN[122X as a [22XG[122X-outer group where [22XG=E/N[122X. A nonabelian cocycle [22Xf: G× G→ N[122X is attached as a component of the [22XG[122X-Outer group.[133X [33X[0;6YThe function can be used without an argument. In this case an empty outer group [22XC[122X is returned. The components must be set using [12XSetActingGroup(C,G)[112X, [12XSetActedGroup(C,N)[112X and [12XSetOuterAction(C,alpha)[112X.[133X │ 
  
  [33X[0;0Y[22XG[122X-cocomplexes[133X
  
      │ CohomologyModule(C,n):: GCocomplex, Int --> GOuterGroup [33X[0;6YInputs a [22XG[122X-cocomplex [22XC[122X together with a non-negative integer [22Xn[122X. It returns the cohomology [22XH^n(C)[122X as a [22XG[122X-outer group. If [22XC[122X was constructed from a [22XZG[122X-resolution [22XR[122X by homing to an abelian [22XG[122X-outer group [22XA[122X then, for each [22Xx[122X in [22XH:=CohomologyModule(C,n)[122X, there is a function [22Xf:=H!.representativeCocycle(x)[122X which is a standard [22Xn[122X-cocycle corresponding to the cohomology class [22Xx[122X. (At present this is implemented only for [22Xn=1,2,3[122X.)[133X │ 
      │ HomToGModule(R,A):: FreeRes, GOuterGroup --> GCocomplex [33X[0;6YInputs a [22XZG[122X-resolution [22XR[122X and an abelian [22XG[122X-outer group [22XA[122X. It returns the [22XG[122X-cocomplex obtained by applying [22XHomZG( _ , A)[122X. (At present this function does not handle equivariant chain maps.)[133X                                                                                                                                                                                                                                         │ 
  
