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`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 resolution`
For 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 complex`
For 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 HE!.phi(n,w) is a function which inputs a non-negative integer n and a word w in B_n(G). It returns the image of w in R_n under a chain equivalence ϕ: B_n(G) → R_n. HE!.psi(n,w) is a function which inputs a non-negative integer n and a word w in R_n. It returns the image of w in B_n(G) under a chain equivalence ψ: R_n → B_n(G). HE!.equiv(n,w) is a function which inputs a non-negative integer n and a word w in B_n(G). It returns the image of w in B_n+1(G) under a ZG-equivariant homomorphism
equiv(n,-) : B_n(G) → B_n+1(G)
satisfyingw - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) . where d(n,-): B_n(G) → B_n-1(G) is the boundary homomorphism in the bar resolution.
This function was implemented by |

`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 HE!.phi(n,w) is a function which inputs a non-negative integer n and a word w in BC_n(G). It returns the image of w in T_n under a chain equivalence ϕ: BC_n(G) → T_n. HE!.psi(n,w) is a function which inputs a non-negative integer n and an element w in T_n. It returns the image of w in BC_n(G) under a chain equivalence ψ: T_n → BC_n(G). HE!.equiv(n,w) is a function which inputs a non-negative integer n and a word w in BC_n(G). It returns the image of w in BC_n+1(G) under a homomorphism
equiv(n,-) : BC_n(G) → BC_n+1(G)
satisfyingw - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) . where d(n,-): BC_n(G) → BC_n-1(G) is the boundary homomorphism in the bar complex.
This function was implemented by |

`Representation of elements in the bar cocomplex`
For 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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