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5 Chain complexes

### 5 Chain complexes

`ChainComplex(T)` Inputs a pure cubical complex, or cubical complex, or simplicial complex T and returns the (often very large) cellular chain complex of T.
`ChainComplexOfPair(T,S)` 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.

`ChevalleyEilenbergComplex(X,n)`

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

`LeibnizComplex(X,n)`

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

`SuspendedChainComplex(C)` 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.
`ReducedSuspendedChainComplex(C)` 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.
`CoreducedChainComplex(C)` `CoreducedChainComplex(C,2)` 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.
`TensorProductOfChainComplexes(C,D)` Inputs two chain complexes C and D of the same characteristic and returns their tensor product as a chain complex. This function was written by Le Van Luyen.
`LefschetzNumber(F)` Inputs a chain map F: C→ C 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).

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