  
  [1X8 [33X[0;0YFunctors[133X[101X
  
  
  [1X8.1 [33X[0;0Y [133X[101X
  
  [1X8.1-1 ExtendScalars[101X
  
  [33X[1;0Y[29X[2XExtendScalars[102X( [3XR[103X, [3XG[103X, [3XEltsG[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  [22XZH[122X-resolution [22XR[122X, a group [22XG[122X containing [22XH[122X as a subgroup, and a list
  [22XEltsG[122X  of  elements of [22XG[122X. It returns the free [22XZG[122X-resolution [22X(R ⊗_ZH ZG)[122X. The
  returned  resolution  [22XS[122X  has  S!.elts:=EltsG.  This  is  a resolution of the
  [22XZG[122X-module [22X(Z ⊗_ZH ZG)[122X. (Here [22X⊗_ZH[122X means tensor over [22XZH[122X.)[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X8.1-2 HomToIntegers[101X
  
  [33X[1;0Y[29X[2XHomToIntegers[102X( [3XX[103X ) [32X function[133X
  
  [33X[0;0YInputs  either  a  [22XZG[122X-resolution [22XX=R[122X, or an equivariant chain map [22XX = (F:R ⟶
  S)[122X.  It  returns  the  cochain  complex  or cochain map obtained by applying
  [22XHomZG( _ , Z)[122X where [22XZ[122X is the trivial module of integers (characteristic 0).[133X
  
  [33X[0;0Y[12XExamples:[112X   1   ([7X../tutorial/chap1.html[107X) ,  2  ([7X../tutorial/chap7.html[107X) ,  3
  ([7X../tutorial/chap8.html[107X) ,       4       ([7X../tutorial/chap10.html[107X) ,       5
  ([7X../tutorial/chap13.html[107X) ,                                                6
  ([7X../www/SideLinks/About/aboutCohomologyRings.html[107X) ,                       7
  ([7X../www/SideLinks/About/aboutSpaceGroup.html[107X) ,                            8
  ([7X../www/SideLinks/About/aboutIntro.html[107X) ,                                 9
  ([7X../www/SideLinks/About/aboutTorAndExt.html[107X) [133X
  
  [1X8.1-3 HomToIntegersModP[101X
  
  [33X[1;0Y[29X[2XHomToIntegersModP[102X( [3XR[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  [22XZG[122X-resolution  [22XR[122X  and  returns  the  cochain  complex obtained by
  applying  [22XHomZG( _ , Z_p)[122X where [22XZ_p[122X is the trivial module of integers mod [22Xp[122X.
  (At present this functor does not handle equivariant chain maps.)[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap8.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutSpaceGroup.html[107X) ,                            3
  ([7X../www/SideLinks/About/aboutIntro.html[107X) ,                                 4
  ([7X../www/SideLinks/About/aboutTorAndExt.html[107X) [133X
  
  [1X8.1-4 HomToIntegralModule[101X
  
  [33X[1;0Y[29X[2XHomToIntegralModule[102X( [3XR[103X, [3Xf[103X ) [32X function[133X
  
  [33X[0;0YInputs a [22XZG[122X-resolution [22XR[122X and a group homomorphism [22Xf:G ⟶ GL_n(Z)[122X to the group
  of  [22Xn×n[122X  invertible  integer matrices. Here [22XZ[122X must have characteristic 0. It
  returns  the  cochain  complex obtained by applying [22XHomZG( _ , A)[122X where [22XA[122X is
  the  [22XZG[122X-module  [22XZ^n[122X  with [22XG[122X action via [22Xf[122X. (At present this function does not
  handle equivariant chain maps.)[133X
  
  [33X[0;0Y[12XExamples:[112X   1  ([7X../tutorial/chap7.html[107X) ,  2  ([7X../tutorial/chap13.html[107X) ,  3
  ([7X../tutorial/chap14.html[107X) ,                                                4
  ([7X../www/SideLinks/About/aboutTwistedCoefficients.html[107X) [133X
  
  [1X8.1-5 TensorWithIntegralModule[101X
  
  [33X[1;0Y[29X[2XTensorWithIntegralModule[102X( [3XR[103X, [3Xf[103X ) [32X function[133X
  
  [33X[0;0YInputs a [22XZG[122X-resolution [22XR[122X and a group homomorphism [22Xf:G ⟶ GL_n(Z)[122X to the group
  of  [22Xn×n[122X  invertible  integer matrices. Here [22XZ[122X must have characteristic 0. It
  returns  the  chain complex obtained by tensoring over [22XZG[122X with the [22XZG[122X-module
  [22XA=Z^n[122X  with  [22XG[122X  action  via  [22Xf[122X.  (At  present  this function does not handle
  equivariant chain maps.)[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../tutorial/chap7.html[107X) [133X
  
  [1X8.1-6 HomToGModule[101X
  
  [33X[1;0Y[29X[2XHomToGModule[102X( [3XR[103X, [3XA[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  [22XZG[122X-resolution  [22XR[122X  and  an abelian G-outer group A. It returns the
  G-cocomplex  obtained  by  applying [22XHomZG( _ , A)[122X. (At present this function
  does not handle equivariant chain maps.)[133X
  
  [33X[0;0Y[12XExamples:[112X   1   ([7X../tutorial/chap6.html[107X) ,  2  ([7X../tutorial/chap7.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutCrossedMods.html[107X) ,                           4
  ([7X../www/SideLinks/About/aboutGouter.html[107X) [133X
  
  [1X8.1-7 InduceScalars[101X
  
  [33X[1;0Y[29X[2XInduceScalars[102X( [3XR[103X, [3Xhom[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  [22XZQ[122X-resolution  [22XR[122X and a surjective group homomorphism [22Xhom:G→ Q[122X. It
  returns the unduced non-free [22XZG[122X-resolution.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X8.1-8 LowerCentralSeriesLieAlgebra[101X
  
  [33X[1;0Y[29X[2XLowerCentralSeriesLieAlgebra[102X( [3XG[103X ) [32X function[133X
  [33X[1;0Y[29X[2XLowerCentralSeriesLieAlgebra[102X( [3Xf[103X ) [32X function[133X
  
  [33X[0;0YInputs a pcp group [22XG[122X. If each quotient [22XG_c/G_c+1[122X of the lower central series
  is  free  abelian  or  p-elementary  abelian  (for fixed prime p) then a Lie
  algebra  [22XL(G)[122X  is  returned. The abelian group underlying [22XL(G)[122X is the direct
  sum  of  the quotients [22XG_c/G_c+1[122X . The Lie bracket on [22XL(G)[122X is induced by the
  commutator in [22XG[122X. (Here [22XG_1=G[122X, [22XG_c+1=[G_c,G][122X .)[133X
  
  [33X[0;0YThe function can also be applied to a group homomorphism [22Xf: G ⟶ G'[122X . In this
  case the induced homomorphism of Lie algebras [22XL(f):L(G) ⟶ L(G')[122X is returned.[133X
  
  [33X[0;0YIf  the  quotients  of  the  lower  central  series  are  not  all  free  or
  p-elementary abelian then the function returns fail.[133X
  
  [33X[0;0YThis function was written by Pablo Fernandez Ascariz[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap7.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutIntro.html[107X) ,                                 3
  ([7X../www/SideLinks/About/aboutLie.html[107X) [133X
  
  [1X8.1-9 TensorWithIntegers[101X
  
  [33X[1;0Y[29X[2XTensorWithIntegers[102X( [3XX[103X ) [32X function[133X
  
  [33X[0;0YInputs  either  a  [22XZG[122X-resolution [22XX=R[122X, or an equivariant chain map [22XX = (F:R ⟶
  S)[122X. It returns the chain complex or chain map obtained by tensoring with the
  trivial module of integers (characteristic 0).[133X
  
  [33X[0;0Y[12XExamples:[112X   1   ([7X../tutorial/chap1.html[107X) ,  2  ([7X../tutorial/chap3.html[107X) ,  3
  ([7X../tutorial/chap6.html[107X) ,        4       ([7X../tutorial/chap7.html[107X) ,       5
  ([7X../tutorial/chap10.html[107X) ,       6       ([7X../tutorial/chap11.html[107X) ,      7
  ([7X../tutorial/chap13.html[107X) ,       8       ([7X../tutorial/chap14.html[107X) ,      9
  ([7X../www/SideLinks/About/aboutArithmetic.html[107X) ,                           10
  ([7X../www/SideLinks/About/aboutArtinGroups.html[107X) ,                          11
  ([7X../www/SideLinks/About/aboutAspherical.html[107X) ,                           12
  ([7X../www/SideLinks/About/aboutParallel.html[107X) ,                             13
  ([7X../www/SideLinks/About/aboutPerformance.html[107X) ,                          14
  ([7X../www/SideLinks/About/aboutCocycles.html[107X) ,                             15
  ([7X../www/SideLinks/About/aboutPersistent.html[107X) ,                           16
  ([7X../www/SideLinks/About/aboutPoincareSeries.html[107X) ,                       17
  ([7X../www/SideLinks/About/aboutCoveringSpaces.html[107X) ,                       18
  ([7X../www/SideLinks/About/aboutCoverinSpaces.html[107X) ,                        19
  ([7X../www/SideLinks/About/aboutPolytopes.html[107X) ,                            20
  ([7X../www/SideLinks/About/aboutCoxeter.html[107X) ,                              21
  ([7X../www/SideLinks/About/aboutRosenbergerMonster.html[107X) ,                   22
  ([7X../www/SideLinks/About/aboutDavisComplex.html[107X) ,                         23
  ([7X../www/SideLinks/About/aboutDefinitions.html[107X) ,                          24
  ([7X../www/SideLinks/About/aboutSimplicialGroups.html[107X) ,                     25
  ([7X../www/SideLinks/About/aboutExtensions.html[107X) ,                           26
  ([7X../www/SideLinks/About/aboutSpaceGroup.html[107X) ,                           27
  ([7X../www/SideLinks/About/aboutFunctorial.html[107X) ,                           28
  ([7X../www/SideLinks/About/aboutGraphsOfGroups.html[107X) ,                       29
  ([7X../www/SideLinks/About/aboutIntro.html[107X) ,                                30
  ([7X../www/SideLinks/About/aboutTorAndExt.html[107X) [133X
  
  [1X8.1-10 FilteredTensorWithIntegers[101X
  
  [33X[1;0Y[29X[2XFilteredTensorWithIntegers[102X( [3XR[103X ) [32X function[133X
  
  [33X[0;0YInputs   a   [22XZG[122X-resolution   [22XR[122X   for   which   "filteredDimension"  lies  in
  NamesOfComponents(R).   (Such   a   resolution   can   be   produced   using
  TwisterTensorProduct(),  ResolutionNormalSubgroups()  or FreeGResolution().)
  It returns the filtered chain complex obtained by tensoring with the trivial
  module of integers (characteristic 0).[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap10.html[107X) ,            2
  ([7X../www/SideLinks/About/aboutPersistent.html[107X) [133X
  
  [1X8.1-11 TensorWithTwistedIntegers[101X
  
  [33X[1;0Y[29X[2XTensorWithTwistedIntegers[102X( [3XX[103X, [3Xrho[103X ) [32X function[133X
  
  [33X[0;0YInputs  either  a  [22XZG[122X-resolution [22XX=R[122X, or an equivariant chain map [22XX = (F:R ⟶
  S)[122X.  It  also  inputs a function [22Xrho: G→ Z[122X where the action of [22Xg ∈ G[122X on [22XZ[122X is
  such  that  [22Xg.1 = rho(g)[122X. It returns the chain complex or chain map obtained
  by tensoring with the (twisted) module of integers (characteristic 0).[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap3.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutCoveringSpaces.html[107X) ,                        3
  ([7X../www/SideLinks/About/aboutCoverinSpaces.html[107X) ,                         4
  ([7X../www/SideLinks/About/aboutTwistedCoefficients.html[107X) [133X
  
  [1X8.1-12 TensorWithIntegersModP[101X
  
  [33X[1;0Y[29X[2XTensorWithIntegersModP[102X( [3XX[103X, [3Xp[103X ) [32X function[133X
  
  [33X[0;0YInputs  either a [22XZG[122X-resolution [22XX=R[122X, or a characteristics 0 chain complex, or
  an   equivariant  chain  map  [22XX  =  (F:R  ⟶  S)[122X,  or  a  chain  map  between
  characteristic  0  chain  complexes, together with a prime [22Xp[122X. It returns the
  chain  complex or chain map obtained by tensoring with the trivial module of
  integers modulo [22Xp[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X   1  ([7X../tutorial/chap1.html[107X) ,  2  ([7X../tutorial/chap10.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutArithmetic.html[107X) ,                            4
  ([7X../www/SideLinks/About/aboutPerformance.html[107X) ,                           5
  ([7X../www/SideLinks/About/aboutPersistent.html[107X) ,                            6
  ([7X../www/SideLinks/About/aboutPoincareSeries.html[107X) ,                        7
  ([7X../www/SideLinks/About/aboutDefinitions.html[107X) ,                           8
  ([7X../www/SideLinks/About/aboutExtensions.html[107X) ,                            9
  ([7X../www/SideLinks/About/aboutTorAndExt.html[107X) [133X
  
  [1X8.1-13 TensorWithTwistedIntegersModP[101X
  
  [33X[1;0Y[29X[2XTensorWithTwistedIntegersModP[102X( [3XX[103X, [3Xp[103X, [3Xrho[103X ) [32X function[133X
  
  [33X[0;0YInputs  either  a  [22XZG[122X-resolution [22XX=R[122X, or an equivariant chain map [22XX = (F:R ⟶
  S)[122X, and a prime [22Xp[122X. It also inputs a function [22Xrho: G→ Z[122X where the action of [22Xg
  ∈  G[122X  on  [22XZ[122X is such that [22Xg.1 = rho(g)[122X. It returns the chain complex or chain
  map obtained by tensoring with the trivial module of integers modulo [22Xp[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutTwistedCoefficients.html[107X) [133X
  
  [1X8.1-14 TensorWithRationals[101X
  
  [33X[1;0Y[29X[2XTensorWithRationals[102X( [3XR[103X ) [32X function[133X
  
  [33X[0;0YInputs a [22XZG[122X-resolution [22XR[122X and returns the chain complex obtained by tensoring
  with the trivial module of rational numbers.[133X
  
  [33X[0;0Y[12XExamples:[112X  1  ([7X../tutorial/chap10.html[107X) ,  2  ([7X../tutorial/chap13.html[107X) ,  3
  ([7X../www/SideLinks/About/aboutExtensions.html[107X) [133X
  
