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, ... , 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 (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 ⟶ 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 ⟶ 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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