GAP (HAP commands) - Chapter 19: Cocycles

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19 Cocycles
19.1
19.1-1 CcGroup
19.1-2 CocycleCondition
19.1-3 StandardCocycle
19.1-4 Syzygy

### 19 Cocycles

#### 19.1

##### 19.1-1 CcGroup

`‣ CcGroup`( A, f ) | ( function )

Inputs a \(G\)-module \(A\) (i.e. an abelian \(G\)-outer group) and a standard 2-cocycle f \(G x G ---> A\). It returns the extension group determined by the cocycle. The group is returned as a CcGroup.

This is a HAPcocyclic function and thus only works when HAPcocyclic is loaded.

Examples: 1 , 2

##### 19.1-2 CocycleCondition

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

Inputs a resolution \(R\) and an integer \(n\)>\(0\). It returns an integer matrix \(M\) with the following property. Suppose \(d=R.dimension(n)\). An integer vector \(f=[f_1, \ldots , f_d]\) then represents a \(ZG\)-homomorphism \(R_n \longrightarrow Z_q\) which sends the \(i\)th generator of \(R_n\) to the integer \(f_i\) in the trivial \(ZG\)-module \(Z_q\) (where possibly \(q=0\) ). The homomorphism \(f\) is a cocycle if and only if \(M^tf=0\) mod \(q\).

Examples: 1

##### 19.1-3 StandardCocycle

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

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

Inputs a \(ZG\)-resolution \(R\) (with contracting homotopy), a positive integer \(n\) and an integer vector \(f\) representing an \(n\)-cocycle \(R_n \longrightarrow Z_q\) 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 \(G_n \longrightarrow Z_q\) corresponding to \(f\). At present the command is implemented only for \(n=2\) or \(3\).

Examples: 1

##### 19.1-4 Syzygy

`‣ Syzygy`( R, g ) | ( function )

Inputs a \(ZG\)-resolution \(R\) (with contracting homotopy) and a list \(g = [g[1], ..., g[n]]\) of elements in \(G\). It returns a word \(w\) in \(R_n\). The word \(w\) is the image of the \(n\)-simplex in the standard bar resolution corresponding to the \(n\)-tuple \(g\). This function can be used to construct explicit standard \(n\)-cocycles. (Currently implemented only for n<4.)

Examples:

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