GAP (HAP commands) - Chapter 9: Chain complexes

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9 Chain complexes
9.1
9.1-1 ChainComplex
9.1-2 ChainComplexOfPair
9.1-3 ChevalleyEilenbergComplex
9.1-4 LeibnizComplex
9.1-5 SuspendedChainComplex
9.1-6 ReducedSuspendedChainComplex
9.1-7 CoreducedChainComplex
9.1-8 TensorProductOfChainComplexes
9.1-9 LefschetzNumber

### 9 Chain complexes

#### 9.1

##### 9.1-1 ChainComplex

`‣ ChainComplex`( T ) | ( function )

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

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11

##### 9.1-2 ChainComplexOfPair

`‣ ChainComplexOfPair`( T, S ) | ( function )

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.

Examples: 1 , 2

##### 9.1-3 ChevalleyEilenbergComplex

`‣ ChevalleyEilenbergComplex`( X, n ) | ( function )

Inputs either a Lie algebra \(X=A\) (over the ring of integers \(Z\) or over a field \(K\)) or a homomorphism of Lie algebras \(X=(f:A \longrightarrow B)\), 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 \(C(f):C(A) \longrightarrow C(B)\).

(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

Examples:

##### 9.1-4 LeibnizComplex

`‣ LeibnizComplex`( X, n ) | ( function )

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 \(X=(f:A \longrightarrow B)\), 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 \(C(f):C(A) \longrightarrow C(B)\).

(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

Examples:

##### 9.1-5 SuspendedChainComplex

`‣ SuspendedChainComplex`( C ) | ( function )

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.

Examples: 1

##### 9.1-6 ReducedSuspendedChainComplex

`‣ ReducedSuspendedChainComplex`( C ) | ( function )

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 \(S_0=\mathbb Z\).

Examples: 1

##### 9.1-7 CoreducedChainComplex

`‣ CoreducedChainComplex`( C ) | ( function )

`‣ CoreducedChainComplex`( C, 2 ) | ( function )

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.

Examples: 1

##### 9.1-8 TensorProductOfChainComplexes

`‣ TensorProductOfChainComplexes`( C, D ) | ( function )

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.

Examples: 1

##### 9.1-9 LefschetzNumber

`‣ LefschetzNumber`( F ) | ( function )

Inputs a chain map \(F\colon C\rightarrow C\) 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).

Examples:

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