GAP (HAP commands) - Chapter 25: Simplicial groups

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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 Simplicial groups

#### 25.1

##### 25.1-1 NerveOfCatOneGroup

`‣ 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.

Examples: 1 , 2 , 3

##### 25.1-2 EilenbergMacLaneSimplicialGroup

`‣ 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.

Examples: 1 , 2 , 3 , 4 , 5

##### 25.1-3 EilenbergMacLaneSimplicialGroupMap

`‣ 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:

##### 25.1-4 MooreComplex

`‣ 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:

##### 25.1-5 ChainComplexOfSimplicialGroup

`‣ 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

##### 25.1-6 SimplicialGroupMap

`‣ 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:

##### 25.1-7 HomotopyGroup

`‣ 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 , 8

##### 25.1-8 Representation of elements in the bar resolution

`‣ 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:

##### 25.1-9 BarResolutionBoundary

`‣ 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:

##### 25.1-10 BarResolutionHomotopy

`‣ 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:

##### 25.1-11 Representation of elements in the bar complex

`‣ 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:

##### 25.1-12 BarComplexBoundary

`‣ 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:

##### 25.1-13 BarResolutionEquivalence

`‣ 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)
  satisfying

  w - \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:

##### 25.1-14 BarComplexEquivalence

`‣ 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)
  satisfying

  w - \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:

##### 25.1-15 Representation of elements in the bar cocomplex

`‣ 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:

##### 25.1-16 BarCocomplexCoboundary

`‣ 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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