‣ 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
‣ 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.
‣ 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
‣ 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:
‣ 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
‣ 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
‣ 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
‣ 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
‣ 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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