Simplicial groups

NerveOfCatOneGroup
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
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
EilenbergMacLaneSimplicialGroupMap(f,n,dim)

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.
MooreComplex
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
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
SimplicialGroupMap(f)

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.
HomotopyGroup
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.

Bar Resolution
Representation of elements in the bar resolution

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}]
.

BarResolutionBoundary
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\colon B_n(G) \rightarrow B_{n-1}(G) in the bar resolution.

This function was implemented by Van Luyen Le.
BarResolutionHomotopy
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\colon B_n(G) \rightarrow B_{n+1}(G) in the bar resolution.

This function is currently being implemented by Van Luyen Le.
Bar Complex
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
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\colon BC_n(G) \rightarrow BC_{n-1}(G) in the bar complex.

This function was implemented by Van Luyen Le.
BarResolutionEquivalence
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 \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.
BarComplexEquivalence
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 \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.
Bar Cocomplex
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
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\colon BC^n(G) \rightarrow BC^{n+1}(G) in the bar cocomplex.
This function was implemented by Van Luyen Le.