GAP (HAP commands) - Chapter 3: Basic functionality for homological group theory

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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 ↣ E ↠ G together with standard 2-cocycle f: G × G → 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 ZG-resolution R of Z and an integer n ≥ 1. It returns an integer matrix M with the following property. Let d be the ZG-rank of R_n. An integer vector f=[f_1, ... , f_d] then represents a ZG-homomorphism R_n → Z_q which sends the ith generator of R_n to the integer f_i in the trivial ZG-module Z_q= Z/q 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 ZG-resolution R (with contracting homotopy), a positive integer n and an integer vector f representing an n-cocycle R_n → Z_q= Z/q Z where G acts trivially on 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 → 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 α: G→ 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 α: G→ 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: G× G→ 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 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 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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