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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<Chapter><Heading> Cocycles</Heading>

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<Table Align="|l|" >

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<Row>

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<Item>

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<Index> CcGroup (HAPcocyclic)</Index>

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<C>CcGroup(A,f) </C>

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<P/>

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Inputs a <M>G</M>-module <M>A</M> (i.e. an abelian <M>G</M>-outer group) and a

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standard 2-cocycle f <M>G x G ---> A</M>. It returns the extension group determined by the cocycle. The group is returned as a CcGroup.

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<P/> This is a HAPcocyclic function and thus only works when HAPcocyclic is loaded.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> CocycleCondition</Index>

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<C>CocycleCondition(R,n) </C>

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<P/>

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Inputs a resolution <M>R</M> and an integer <M>n</M>&tgt;<M>0</M>. 

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It returns an integer matrix <M>M</M> 

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with the following property. Suppose <M>d=R.dimension(n)</M>. 

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An integer vector <M>f=[f_1, \ldots , f_d]</M> 

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then represents a <M>ZG</M>-homomorphism <M>R_n \longrightarrow Z_q</M> 

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which sends the <M>i</M>th generator of <M>R_n</M> to the integer 

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<M>f_i</M> in the trivial <M>ZG</M>-module <M>Z_q</M> (where possibly <M>q=0</M>

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). The homomorphism <M>f</M> is a cocycle if and only if <M>M^tf=0</M>

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mod <M>q</M>. 

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> StandardCocycle</Index>

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<C>StandardCocycle(R,f,n) </C>

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<Br/>

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<C>StandardCocycle(R,f,n,q) </C>

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<P/>

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Inputs a <M>ZG</M>-resolution <M>R</M> (with contracting homotopy), 

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a positive integer <M>n</M> and an integer vector <M>f</M> 

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representing an <M>n</M>-cocycle <M>R_n \longrightarrow Z_q</M> 

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where <M>G</M> acts trivially on <M>Z_q</M>. It is assumed <M>q=0</M>

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unless a value for <M>q</M> is entered. The command returns a function 

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<M>F(g_1, ..., g_n)</M> 

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which is the standard cocycle  <M>G_n \longrightarrow Z_q</M>

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corresponding to <M>f</M>. At present the command is implemented only for 

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<M>n=2</M> or <M>3</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> Syzygy</Index>

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<C>Syzygy(R,g) </C>

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<P/>

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Inputs a <M>ZG</M>-resolution <M>R</M> 

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(with contracting homotopy) and a list 

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<M>g = [g[1], ..., g[n]]</M> of elements in <M>G</M>. 

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It returns a word <M>w</M> in <M>R_n</M>. 

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The word <M>w</M> is the image of the <M>n</M>-simplex in the 

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standard bar resolution corresponding to the <M>n</M>-tuple <M>g</M>. 

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This function can be used to construct explicit standard <M>n</M>-cocycles. 

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(Currently implemented only for n&tlt;4.)

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</Item>

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</Row>

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</Table>

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</Chapter>

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