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

##### 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 \(n-1\)st 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

##### 25.1-3 EilenbergMacLaneSimplicialGroupMap

`‣ EilenbergMacLaneSimplicialGroupMap` | ( global variable )

Inputs a group homomorphism \(f:G\rightarrow Q\), a positive integer \(n\), and a positive integer \(dim \). The function returns the first \(1+dim\) terms of a simplicial group homomorphism \(f:K(G,n) \rightarrow K(Q,n)\) 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

##### 25.1-6 SimplicialGroupMap

`‣ SimplicialGroupMap` | ( global variable )

Inputs a homomorphism \(f:G\rightarrow Q\) of simplicial groups. The function returns an induced map \(f:C(G) \rightarrow C(Q)\) 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

##### 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 \(\mathbb ZG\)-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 \(d_n\colon B_n(G) \rightarrow B_{n-1}(G)\) 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 \(h_n\colon B_n(G) \rightarrow B_{n+1}(G)\) 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 \(d_n\colon BC_n(G) \rightarrow BC_{n-1}(G)\) 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 \(\phi\colon B_n(G) \rightarrow R_n\).

- 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 \(\psi\colon R_n \rightarrow B_n(G)\).

- 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
  \(equiv(n,-) \colon B_n(G) \rightarrow B_{n+1}(G)\)
  satisfying

  \[w - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) . \]

  where \(d(n,-)\colon B_n(G) \rightarrow B_{n-1}(G)\) 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 \(\phi\colon BC_n(G) \rightarrow T_n\).

- 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 \(\psi\colon T_n \rightarrow BC_n(G)\).

- 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
  \(equiv(n,-) \colon BC_n(G) \rightarrow BC_{n+1}(G)\)
  satisfying

  \[w - \psi ( \phi (w)) = d(n+1, equiv(n,w)) + equiv(n-1,d(n,w)) . \]

  where \(d(n,-)\colon BC_n(G) \rightarrow BC_{n-1}(G)\) 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 \(d^n\colon BC^n(G) \rightarrow BC^{n+1}(G)\) in the bar cocomplex.
This function was implemented by Van Luyen Le.

Examples:

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