  
  [1X9 [33X[0;0YChain complexes[133X[101X
  
  
  [1X9.1 [33X[0;0Y [133X[101X
  
  [1X9.1-1 ChainComplex[101X
  
  [33X[1;0Y[29X[2XChainComplex[102X( [3XT[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  pure cubical complex, or cubical complex, or simplicial complex [22XT[122X
  and returns the (often very large) cellular chain complex of [22XT[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X   1   ([7X../tutorial/chap1.html[107X) ,  2  ([7X../tutorial/chap3.html[107X) ,  3
  ([7X../tutorial/chap10.html[107X) , 4 ([7X../www/SideLinks/About/aboutMetrics.html[107X) , 5
  ([7X../www/SideLinks/About/aboutBredon.html[107X) ,                                6
  ([7X../www/SideLinks/About/aboutPersistent.html[107X) ,                            7
  ([7X../www/SideLinks/About/aboutCoveringSpaces.html[107X) ,                        8
  ([7X../www/SideLinks/About/aboutCoverinSpaces.html[107X) ,                         9
  ([7X../www/SideLinks/About/aboutCubical.html[107X) ,                              10
  ([7X../www/SideLinks/About/aboutSimplicialGroups.html[107X) ,                     11
  ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X9.1-2 ChainComplexOfPair[101X
  
  [33X[1;0Y[29X[2XChainComplexOfPair[102X( [3XT[103X, [3XS[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  pure  cubical  complex  or  cubical  complex  [22XT[122X  and contractible
  subcomplex [22XS[122X. It returns the quotient [22XC(T)/C(S)[122X of cellular chain complexes.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap10.html[107X) ,            2
  ([7X../www/SideLinks/About/aboutCubical.html[107X) [133X
  
  [1X9.1-3 ChevalleyEilenbergComplex[101X
  
  [33X[1;0Y[29X[2XChevalleyEilenbergComplex[102X( [3XX[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs either a Lie algebra [22XX=A[122X (over the ring of integers [22XZ[122X or over a field
  [22XK[122X)  or  a homomorphism of Lie algebras [22XX=(f:A ⟶ B)[122X, together with a positive
  integer  [22Xn[122X.  It  returns either the first [22Xn[122X terms of the Chevalley-Eilenberg
  chain  complex  [22XC(A)[122X,  or  the  induced map of Chevalley-Eilenberg complexes
  [22XC(f):C(A) ⟶ C(B)[122X.[133X
  
  [33X[0;0Y(The  homology  of the Chevalley-Eilenberg complex [22XC(A)[122X is by definition the
  homology of the Lie algebra [22XA[122X with trivial coefficients in [22XZ[122X or [22XK[122X).[133X
  
  [33X[0;0YThis function was written by [12XPablo Fernandez Ascariz[112X[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X9.1-4 LeibnizComplex[101X
  
  [33X[1;0Y[29X[2XLeibnizComplex[102X( [3XX[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs  either  a Lie or Leibniz algebra [22XX=A[122X (over the ring of integers [22XZ[122X or
  over  a  field  [22XK[122X) or a homomorphism of Lie or Leibniz algebras [22XX=(f:A ⟶ B)[122X,
  together  with  a positive integer [22Xn[122X. It returns either the first [22Xn[122X terms of
  the  Leibniz  chain  complex  [22XC(A)[122X,  or the induced map of Leibniz complexes
  [22XC(f):C(A) ⟶ C(B)[122X.[133X
  
  [33X[0;0Y(The  Leibniz  complex  [22XC(A)[122X  was  defined by J.-L.Loday. Its homology is by
  definition the Leibniz homology of the algebra [22XA[122X).[133X
  
  [33X[0;0YThis function was written by [12XPablo Fernandez Ascariz[112X[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X9.1-5 SuspendedChainComplex[101X
  
  [33X[1;0Y[29X[2XSuspendedChainComplex[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0YInputs a chain complex [22XC[122X and returns the chain complex [22XS[122X defined by applying
  the degree shift [22XS_n = C_n-1[122X to chain groups and boundary homomorphisms.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../tutorial/chap10.html[107X) [133X
  
  [1X9.1-6 ReducedSuspendedChainComplex[101X
  
  [33X[1;0Y[29X[2XReducedSuspendedChainComplex[102X( [3XC[103X ) [32X function[133X
  
  [33X[0;0YInputs a chain complex [22XC[122X and returns the chain complex [22XS[122X defined by applying
  the  degree shift [22XS_n = C_n-1[122X to chain groups and boundary homomorphisms for
  all [22Xn > 0[122X. The chain complex [22XS[122X has trivial homology in degree [22X0[122X and [22XS_0= Z[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../tutorial/chap10.html[107X) [133X
  
  [1X9.1-7 CoreducedChainComplex[101X
  
  [33X[1;0Y[29X[2XCoreducedChainComplex[102X( [3XC[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCoreducedChainComplex[102X( [3XC[103X, [3X2[103X ) [32X function[133X
  
  [33X[0;0YInputs  a chain complex [22XC[122X and returns a quasi-isomorphic chain complex [22XD[122X. In
  many  cases  the  complex  [22XD[122X should be smaller than [22XC[122X. If an optional second
  input  argument  is  set  equal  to 2 then an alternative method is used for
  reducing the size of the chain complex.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutSimplicialGroups.html[107X) [133X
  
  [1X9.1-8 TensorProductOfChainComplexes[101X
  
  [33X[1;0Y[29X[2XTensorProductOfChainComplexes[102X( [3XC[103X, [3XD[103X ) [32X function[133X
  
  [33X[0;0YInputs  two  chain  complexes [22XC[122X and [22XD[122X of the same characteristic and returns
  their tensor product as a chain complex.[133X
  
  [33X[0;0YThis function was written by [12X Le Van Luyen[112X.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutSimplicialGroups.html[107X) [133X
  
  [1X9.1-9 LefschetzNumber[101X
  
  [33X[1;0Y[29X[2XLefschetzNumber[102X( [3XF[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  chain  map  [22XF: C→ C[122X with common source and target. It returns the
  Lefschetz  number  of the map (that is, the alternating sum of the traces of
  the homology maps in each degree).[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
