  
  [1X5. Chain complexes[0X
  
      | [10XChainComplex(T)[0X Inputs a pure cubical complex, or cubical complex, or simplicial complex T and returns the (often very large) cellular chain complex of T.                                                                                                                                                                                                                                                                                                                                                                                                           | 
      | [10XChainComplexOfPair(T,S)[0X Inputs a pure cubical complex or cubical complex T and contractible subcomplex S. It returns the quotient C(T)/C(S) of cellular chain complexes.                                                                                                                                                                                                                                                                                                                                                                                             | 
      | [10X ChevalleyEilenbergComplex(X,n) [0X Inputs either a Lie algebra X=A (over the ring of integers Z or over a field K) or a homomorphism of Lie algebras X=(f:A --> B), together with a positive integer n. It returns either the first n terms of the Chevalley-Eilenberg chain complex C(A), or the induced map of Chevalley-Eilenberg complexes C(f):C(A) --> C(B). (The homology of the Chevalley-Eilenberg complex C(A) is by definition the homology of the Lie algebra A with trivial coefficients in Z or K). This function was written by Pablo Fernandez Ascariz | 
      | [10X LeibnizComplex(X,n) [0X Inputs either a Lie or Leibniz algebra X=A (over the ring of integers Z or over a field K) or a homomorphism of Lie or Leibniz algebras X=(f:A --> B), together with a positive integer n. It returns either the first n terms of the Leibniz chain complex C(A), or the induced map of Leibniz complexes C(f):C(A) --> C(B). (The Leibniz complex C(A) was defined by J.-L.Loday. Its homology is by definition the Leibniz homology of the algebra A). This function was written by Pablo Fernandez Ascariz                                  | 
      | [10XSuspendedChainComplex(C)[0X Inputs a chain complex C and returns the chain complex S defined by applying the degree shift S_n = C_n-1 to chain groups and boundary homomorphisms.                                                                                                                                                                                                                                                                                                                                                                                       | 
      | [10XReducedSuspendedChainComplex(C)[0X Inputs a chain complex C and returns the chain complex S defined by applying the degree shift S_n = C_n-1 to chain groups and boundary homomorphisms for all n > 0. The chain complex S has trivial homology in degree 0 and S_0= Z.                                                                                                                                                                                                                                                                                                 | 
      | [10XCoreducedChainComplex(C)[0X [10XCoreducedChainComplex(C,2)[0X Inputs a chain complex C and returns a quasi-isomorphic chain complex D. In many cases the complex D should be smaller than C. 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.                                                                                                                                                                                                                                             | 
  
