  
  [1X19 [33X[0;0YCocycles[133X[101X
  
  
  [1X19.1 [33X[0;0Y [133X[101X
  
  [1X19.1-1 CcGroup[101X
  
  [33X[1;0Y[29X[2XCcGroup[102X( [3XA[103X, [3Xf[103X ) [32X function[133X
  
  [33X[0;0YInputs a [22XG[122X-module [22XA[122X (i.e. an abelian [22XG[122X-outer group) and a standard 2-cocycle
  f  [22XG  x  G ---> A[122X. It returns the extension group determined by the cocycle.
  The group is returned as a CcGroup.[133X
  
  [33X[0;0YThis  is  a  HAPcocyclic  function  and  thus only works when HAPcocyclic is
  loaded.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap5.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutGouter.html[107X) [133X
  
  [1X19.1-2 CocycleCondition[101X
  
  [33X[1;0Y[29X[2XCocycleCondition[102X( [3XR[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  resolution  [22XR[122X  and an integer [22Xn[122X>[22X0[122X. It returns an integer matrix [22XM[122X
  with  the  following  property.  Suppose [22Xd=R.dimension(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[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
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutCocycles.html[107X) [133X
  
  [1X19.1-3 StandardCocycle[101X
  
  [33X[1;0Y[29X[2XStandardCocycle[102X( [3XR[103X, [3Xf[103X, [3Xn[103X ) [32X function[133X
  [33X[1;0Y[29X[2XStandardCocycle[102X( [3XR[103X, [3Xf[103X, [3Xn[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  [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[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;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutCocycles.html[107X) [133X
  
  [1X19.1-4 Syzygy[101X
  
  [33X[1;0Y[29X[2XSyzygy[102X( [3XR[103X, [3Xg[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  [22XZG[122X-resolution [22XR[122X (with contracting homotopy) and a list [22Xg = [g[1],
  ...,  g[n]][122X  of elements in [22XG[122X. It returns a word [22Xw[122X in [22XR_n[122X. The word [22Xw[122X is the
  image  of  the [22Xn[122X-simplex in the standard bar resolution corresponding to the
  [22Xn[122X-tuple  [22Xg[122X.  This  function  can  be  used  to  construct  explicit standard
  [22Xn[122X-cocycles. (Currently implemented only for n<4.)[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
