| NerveOfCatOneGroup(G,n)Inputs a cat-1-group G and a positive integer n. It returns the low-dimensional part of the nerve of G as a simplicial group of length n.  | 
| EilenbergMacLaneSimplicialGroup(G,n,dim)Inputs a group G, a positive integer n, and a positive integer dim. The function returns the first 1+dim terms of a simplicial group with n-1st homotopy group equal to G and all other homotopy groups equal to zero.  | 
| EilenbergMacLaneSimplicialGroupMap(f,n,dim)Inputs a group homomorphism f:G→ Q, a positive integer n, and a positive integer dim. The function returns the first 1+dim terms of a simplicial group homomorphism f:K(G,n) → K(Q,n) of Eilenberg-MacLane simplicial groups.  | 
| MooreComplex(G)Inputs a simplicial group G and returns its Moore complex as a G-complex.  | 
| ChainComplexOfSimplicialGroup(G)Inputs a simplicial group G and returns the cellular chain complex C of a CW-space X represented by the homotopy type of the simplicial group. Thus the homology groups of C are the integral homology groups of X.  | 
| SimplicialGroupMap(f)Inputs a homomorphism f:G→ Q of simplicial groups. The function returns an induced map f:C(G) → C(Q) of chain complexes whose homology is the integral homology of the simplicial group G and Q respectively.  | 
| HomotopyGroup(G,n)Inputs a simplicial group G and a positive integer n. The integer n must be less than the length of G. It returns, as a group, the (n)-th homology group of its Moore complex. Thus HomotopyGroup(G,0) returns the "fundamental group" of G. | 
| Representation of elements in the bar resolutionFor a group G we denote by B_n(G) the free ZG-module with basis the lists [g_1 | g_2 | ... | g_n] where the g_i range over G.  | 
| BarResolutionBoundary(w)This function inputs a word w in the bar resolution module B_n(G) and returns its image under the boundary homomorphism d_n: B_n(G) → B_n-1(G) in the bar resolution.  | 
| BarResolutionHomotopy(w)This function inputs a word w in the bar resolution module B_n(G) and returns its image under the contracting homotopy h_n: B_n(G) → B_n+1(G) in the bar resolution.  | 
| Representation of elements in the bar complexFor a group G we denote by BC_n(G) the free abelian group with basis the lists [g_1 | g_2 | ... | g_n] where the g_i range over G.  | 
| BarComplexBoundary(w)This function inputs a word w in the n-th term of the bar complex BC_n(G) and returns its image under the boundary homomorphism d_n: BC_n(G) → BC_n-1(G) in the bar complex.  | 
| BarResolutionEquivalence(R)This function inputs a free ZG-resolution R. It returns a component object HE with components 
 This function was implemented by Van Luyen Le. | 
| BarComplexEquivalence(R)This function inputs a free ZG-resolution R. It first constructs the chain complex T=TensorWithIntegerts(R). The function returns a component object HE with components 
 This function was implemented by Van Luyen Le. | 
| Representation of elements in the bar cocomplexFor a group G we denote by BC^n(G) the free abelian group with basis the lists [g_1 | g_2 | ... | g_n] where the g_i range over G.  | 
| BarCocomplexCoboundary(w)This function inputs a word w in the n-th term of the bar cocomplex BC^n(G) and returns its image under the coboundary homomorphism d^n: BC^n(G) → BC^n+1(G) in the bar cocomplex.  | 
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