HAP home
 Top of Book   Previous Chapter   Next Chapter 

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.


 


 Top of Book   Previous Chapter   Next Chapter 
Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 Ind

generated by GAPDoc2HTML