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19 Cocycles

19.1  

19.1-1 CcGroup

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

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 , 2 

19.1-3 StandardCocycle

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 , 2 

19.1-4 Syzygy

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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