<Chapter><Heading> Cocycles</Heading>

<Table Align="|l|" >

<Row>
<Item>
<Index> CcGroup (HAPcocyclic)</Index>
<C>
CcGroup(A,f)
</C>
<P/>

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

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

</Item>
</Row>


<Row>
<Item>
<Index> CocycleCondition</Index>
<C> 
CocycleCondition(R,n)
</C>
<P/>

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

<Row>
<Item>
<Index> StandardCocycle</Index>
<C>
StandardCocycle(R,f,n)
</C>
<Br/>
<C>
StandardCocycle(R,f,n,q)
</C>
<P/>

Inputs a <M>ZG</M>-resolution <M>R</M> (with contracting homotopy), 
a positive integer <M>n</M> and an integer vector <M>f</M> 
representing an <M>n</M>-cocycle <M>R_n \longrightarrow Z_q</M> 
where <M>G</M> acts trivially on <M>Z_q</M>. It is assumed <M>q=0</M>
unless a value for <M>q</M> is entered. The command returns a function 
<M>F(g_1, ..., g_n)</M> 
which is the standard cocycle  <M>G_n \longrightarrow Z_q</M>
corresponding to <M>f</M>. At present the command is implemented only for 
<M>n=2</M> or <M>3</M>.
</Item>
</Row>

<Row>
<Item>
<Index> Syzygy</Index>
<C>
Syzygy(R,g)
</C>
<P/>

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

</Table>
</Chapter>