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21 Simplicial groups

### 21 Simplicial groups

`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. 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)` 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(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. This function was implemented by Van Luyen Le.
`MooreComplex(G)` Inputs a simplicial group G and returns its Moore complex as a G-complex. This function was implemented by Van Luyen Le.
`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. This function was implemented by Van Luyen Le.
`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. This function was implemented by Van Luyen Le.
`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.
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].

`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. This function was implemented by Van Luyen Le.
`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. This function is currently being implemented by Van Luyen Le.

`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.
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].

`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. This function was implemented by Van Luyen Le.

`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)
  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.

`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)
  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.

`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.
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].

`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. This function was implemented by Van Luyen Le.

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