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3 Basic functionality for homological group theory
3.1 Cocycles
3.1-1 CcGroup
3.1-2 CocycleCondition
3.1-3 StandardCocycle
3.2 G-Outer Groups
3.2-1 ActedGroup
3.2-2 ActingGroup
3.2-3 Centre
3.2-4 GOuterGroup
3.3 \(G\)-cocomplexes
3.3-1 CohomologyModule
3.3-2 HomToGModule

### 3 Basic functionality for homological group theory

This page covers the functions used in chapter 4 of the book An Invitation to Computational Homotopy.

#### 3.1 Cocycles

##### 3.1-1 CcGroup

`‣ CcGroup`( N, f ) | ( function )

Inputs a \(G\)-outer group \(N\) with nonabelian cocycle describing some extension \(N \rightarrowtail E \twoheadrightarrow G\) together with standard 2-cocycle \(f\colon G \times G \rightarrow A\) where \(A=Z(N)\). It returns the extension group determined by the cocycle \(f\). The group is returned as a cocyclic group.

This function is part of the HAPcocyclic package of functions implemented by Robert F. Morse.

Examples: 1 , 2

##### 3.1-2 CocycleCondition

`‣ CocycleCondition`( R, n ) | ( function )

Inputs a free \(\mathbb ZG\)-resolution \(R\) of \(\mathbb Z\) and an integer \(n \ge 1\). It returns an integer matrix \(M\) with the following property. Let \(d\) be the \(\mathbb ZG\)-rank of \(R_n\). An integer vector \(f=[f_1, ... , f_d]\) then represents a \(\mathbb ZG\)-homomorphism \(R_n \rightarrow \mathbb Z_q\) which sends the \(i\)th generator of \(R_n\) to the integer \(f_i\) in the trivial \(\mathbb ZG\)-module \(\mathbb Z_q=\mathbb Z/q{\mathbb Z}\) (where possibly \(q=0\)). The homomorphism \(f\) is a cocycle if and only if \(M^tf=0\) mod \(q\).

Examples: 1 , 2

##### 3.1-3 StandardCocycle

`‣ StandardCocycle`( R, f, n ) | ( function )

`‣ StandardCocycle`( R, f, n, q ) | ( function )

Inputs a free \(\mathbb ZG\)-resolution \(R\) (with contracting homotopy), a positive integer \(n\) and an integer vector \(f\) representing an \(n\)-cocycle \(R_n \rightarrow \mathbb Z_q=\mathbb Z/q\mathbb Z\) where \(G\) acts trivially on \(\mathbb Z_q\). It is assumed \(q=0\) unless a value for \(q\) is entered. The command returns a function \(F(g_1, ..., g_n)\) which is the standard cocycle \(G^n \rightarrow \mathbb Z_q\) corresponding to \(f\). At present the command is implemented only for \(n=2\) or \(3\).

Examples: 1 , 2

#### 3.2 G-Outer Groups

##### 3.2-1 ActedGroup

`‣ ActedGroup`( M ) | ( function )

Inputs a \(G\)-outer group \(M\) corresponding to a homomorphism \(\alpha\colon G\rightarrow {\rm Out}(N)\) and returns the group \(N\).

Examples: 1 , 2 , 3 , 4

##### 3.2-2 ActingGroup

`‣ ActingGroup`( M ) | ( function )

Inputs a \(G\)-outer group \(M\) corresponding to a homomorphism \(\alpha\colon G\rightarrow {\rm Out}(N)\) and returns the group \(G\).

Examples: 1 , 2

##### 3.2-3 Centre

`‣ Centre`( M ) | ( function )

Inputs a \(G\)-outer group \(M\) and returns its group-theoretic centre as a \(G\)-outer group.

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

##### 3.2-4 GOuterGroup

`‣ GOuterGroup`( E, N ) | ( function )

`‣ GOuterGroup`( ) | ( function )

Inputs a group \(E\) and normal subgroup \(N\). It returns \(N\) as a \(G\)-outer group where \(G=E/N\). A nonabelian cocycle \(f\colon G\times G\rightarrow N\) is attached as a component of the \(G\)-Outer group.

The function can be used without an argument. In this case an empty outer group \(C\) is returned. The components must be set using SetActingGroup(C,G), SetActedGroup(C,N) and SetOuterAction(C,alpha).

Examples: 1 , 2 , 3 , 4

#### 3.3 \(G\)-cocomplexes

##### 3.3-1 CohomologyModule

`‣ CohomologyModule`( C, n ) | ( function )

Inputs a \(G\)-cocomplex \(C\) together with a non-negative integer \(n\). It returns the cohomology \(H^n(C)\) as a \(G\)-outer group. If \(C\) was constructed from a \(\mathbb ZG\)-resolution \(R\) by homing to an abelian \(G\)-outer group \(A\) then, for each \(x\) in \(H:=CohomologyModule(C,n)\), there is a function \(f:=H!.representativeCocycle(x)\) which is a standard \(n\)-cocycle corresponding to the cohomology class \(x\). (At present this is implemented only for \(n=1,2,3\).)

Examples: 1 , 2 , 3

##### 3.3-2 HomToGModule

`‣ HomToGModule`( R, A ) | ( function )

Inputs a \(\mathbb ZG\)-resolution \(R\) and an abelian \(G\)-outer group \(A\). It returns the \(G\)-cocomplex obtained by applying \(HomZG( \_ , A)\). (At present this function does not handle equivariant chain maps.)

Examples: 1 , 2 , 3 , 4

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