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 , 12 , 13

##### 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 ⟶ 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

Examples: 1

##### 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 ⟶ 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

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= 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: 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).

Examples:

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