GAP (HAP) - Chapter 3: Homological Group Theory

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3 Homological Group Theory

### 3 Homological Group Theory

Cocycles

    CcGroup(N,f):: GOuterGroup, StandardCocycle --> CcGroup
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.
    CocycleCondition(R,n):: FreeRes, Int --> IntMat
Inputs a free ZG-resolution R of Z and an integer n ge 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.
    StandardCocycle(R,f,n):: FreeRes, List, Int --> Function
    StandardCocycle(R,f,n,q):: FreeRes, List, Int --> 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.

G-Outer Groups

    ActedGroup(M):: GOuterGroup --> Group
Inputs a G-outer group M corresponding to a homomorphism α: G→ Out(N) and returns the group $N$.
    ActingGroup(M):: GOuterGroup --> Group
Inputs a G-outer group M corresponding to a homomorphism α: G→ Out(N) and returns the group $G$.
    Centre(M):: GOuterGroup --> GOuterGroup
Inputs a G-outer group M and returns its group-theoretic centre as a G-outer group.
    GOuterGroup(E,N):: Group, Subgroup --> GOuterGroup
    GOuterGroup():: Group, Subgroup --> GOuterGroup
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).

G-cocomplexes

    CohomologyModule(C,n):: GCocomplex, Int --> GOuterGroup
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.)
    HomToGModule(R,A):: FreeRes, GOuterGroup --> GCocomplex
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.)

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