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<Chapter><Heading> Cocycles</Heading> <Section><Heading> </Heading>
<ManSection> <Func Name="CcGroup" Arg="A,f"/> <Description> <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. <P/><B>Examples:</B> <URL><Link>../tutorial/chap6.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutGouter.html</Link><LinkText>2</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="CocycleCondition" Arg="R,n"/> <Description> <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>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap7.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCocycles.html</Link><LinkText>2</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="StandardCocycle" Arg="R,f,n"/> <Func Name="StandardCocycle" Arg="R,f,n,q"/> <Description> <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>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap7.html</Link><LinkText>1</LinkText></URL> , <URL><Link>../www/SideLinks/About/aboutCocycles.html</Link><LinkText>2</LinkText></URL>
</Description> </ManSection>
<ManSection> <Func Name="Syzygy" Arg="R,g"/> <Description> <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.) <P/><B>Examples:</B>
</Description> </ManSection> </Section> </Chapter>
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