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25 Simplicial groups

25.1

25.1-1 NerveOfCatOneGroup

25.1-2 EilenbergMacLaneSimplicialGroup

25.1-3 EilenbergMacLaneSimplicialGroupMap

25.1-4 MooreComplex

25.1-5 ChainComplexOfSimplicialGroup

25.1-6 SimplicialGroupMap

25.1-7 HomotopyGroup

25.1-8 Representation of elements in the bar resolution

25.1-9 BarResolutionBoundary

25.1-10 BarResolutionHomotopy

25.1-11 Representation of elements in the bar complex

25.1-12 BarComplexBoundary

25.1-13 BarResolutionEquivalence

25.1-14 BarComplexEquivalence

25.1-15 Representation of elements in the bar cocomplex

25.1-16 BarCocomplexCoboundary

25.1-1 NerveOfCatOneGroup

25.1-2 EilenbergMacLaneSimplicialGroup

25.1-3 EilenbergMacLaneSimplicialGroupMap

25.1-4 MooreComplex

25.1-5 ChainComplexOfSimplicialGroup

25.1-6 SimplicialGroupMap

25.1-7 HomotopyGroup

25.1-8 Representation of elements in the bar resolution

25.1-9 BarResolutionBoundary

25.1-10 BarResolutionHomotopy

25.1-11 Representation of elements in the bar complex

25.1-12 BarComplexBoundary

25.1-13 BarResolutionEquivalence

25.1-14 BarComplexEquivalence

25.1-15 Representation of elements in the bar cocomplex

25.1-16 BarCocomplexCoboundary

`‣ NerveOfCatOneGroup` ( G, n ) | ( function ) |

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.

This function applies both to cat-1-groups for which IsHapCatOneGroup(G) is true, and to cat-1-groups produced using the Xmod package.

This function was implemented by **Van Luyen Le**.

`‣ EilenbergMacLaneSimplicialGroup` ( G, n, dim ) | ( function ) |

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.

This function was implemented by **Van Luyen Le**.

`‣ EilenbergMacLaneSimplicialGroupMap` | ( global variable ) |

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.

This function was implemented by **Van Luyen Le**.

**Examples:**

`‣ MooreComplex` ( G ) | ( function ) |

Inputs a simplicial group G and returns its Moore complex as a G-complex.

This function was implemented by **Van Luyen Le**.

**Examples:**

`‣ ChainComplexOfSimplicialGroup` ( G ) | ( function ) |

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.

This function was implemented by **Van Luyen Le**.

**Examples:** 1 , 2 , 3 , 4 , 5 , 6

`‣ SimplicialGroupMap` | ( global variable ) |

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.

This function was implemented by **Van Luyen Le**.

**Examples:**

`‣ HomotopyGroup` ( G, n ) | ( function ) |

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.

**Examples:** 1 , 2 , 3 , 4 , 5 , 6 , 7

`‣ Representation of elements in the bar resolution` | ( global variable ) |

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.

We represent a word

w = h_1.[g_11 | g_12 | ... | g_1n] - h_2.[g_21 | g_22 | ... | g_2n] + ... + h_k.[g_k1 | g_k2 | ... | g_kn]

in B_n(G) as a list of lists:

[ [+1,h_1,g_11 , g_12 , ... , g_1n] , [-1, h_2,g_21 , g_22 , ... | g_2n] + ... + [+1, h_k,g_k1 , g_k2 , ... , g_kn].

**Examples:**

`‣ BarResolutionBoundary` | ( global variable ) |

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.

This function was implemented by **Van Luyen Le**.

**Examples:**

`‣ BarResolutionHomotopy` | ( global variable ) |

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.

This function is currently being implemented by **Van Luyen Le**.

**Examples:**

`‣ Representation of elements in the bar complex` | ( global variable ) |

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.

We represent a word

w = [g_11 | g_12 | ... | g_1n] - [g_21 | g_22 | ... | g_2n] + ... + [g_k1 | g_k2 | ... | g_kn]

in BC_n(G) as a list of lists:

[ [+1,g_11 , g_12 , ... , g_1n] , [-1, g_21 , g_22 , ... | g_2n] + ... + [+1, g_k1 , g_k2 , ... , g_kn].

**Examples:**

`‣ BarComplexBoundary` | ( global variable ) |

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.

This function was implemented by **Van Luyen Le**.

**Examples:**

`‣ BarResolutionEquivalence` ( R ) | ( function ) |

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 **Van Luyen Le**.

**Examples:**

`‣ BarComplexEquivalence` ( R ) | ( function ) |

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 **Van Luyen Le**.

**Examples:**

`‣ Representation of elements in the bar cocomplex` | ( global variable ) |

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.

We represent a word

w = [g_11 | g_12 | ... | g_1n] - [g_21 | g_22 | ... | g_2n] + ... + [g_k1 | g_k2 | ... | g_kn]

in BC^n(G) as a list of lists:

[ [+1,g_11 , g_12 , ... , g_1n] , [-1, g_21 , g_22 , ... | g_2n] + ... + [+1, g_k1 , g_k2 , ... , g_kn].

**Examples:**

`‣ BarCocomplexCoboundary` | ( global variable ) |

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.

This function was implemented by **Van Luyen Le**.

**Examples:**

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