<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CcGroup</code>( <var class="Arg">A</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">G</span>-module <span class="SimpleMath">A</span> (i.e. an abelian <span class="SimpleMath">G</span>-outer group) and a standard 2-cocycle f <span class="SimpleMath">G x G ---> A</span>. It returns the extension group determined by the cocycle. The group is returned as a CcGroup.</p>
<p>This is a HAPcocyclic function and thus only works when HAPcocyclic is loaded.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CocycleCondition</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a resolution <span class="SimpleMath">R</span> and an integer <span class="SimpleMath">n</span>><span class="SimpleMath">0</span>. It returns an integer matrix <span class="SimpleMath">M</span> with the following property. Suppose <span class="SimpleMath">d=R.dimension(n)</span>. An integer vector <span class="SimpleMath">f=[f_1, ... , f_d]</span> then represents a <span class="SimpleMath">ZG</span>-homomorphism <span class="SimpleMath">R_n ⟶ Z_q</span> which sends the <span class="SimpleMath">i</span>th generator of <span class="SimpleMath">R_n</span> to the integer <span class="SimpleMath">f_i</span> in the trivial <span class="SimpleMath">ZG</span>-module <span class="SimpleMath">Z_q</span> (where possibly <span class="SimpleMath">q=0</span> ). The homomorphism <span class="SimpleMath">f</span> is a cocycle if and only if <span class="SimpleMath">M^tf=0</span> mod <span class="SimpleMath">q</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardCocycle</code>( <var class="Arg">R</var>, <var class="Arg">f</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardCocycle</code>( <var class="Arg">R</var>, <var class="Arg">f</var>, <var class="Arg">n</var>, <var class="Arg">q</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> (with contracting homotopy), a positive integer <span class="SimpleMath">n</span> and an integer vector <span class="SimpleMath">f</span> representing an <span class="SimpleMath">n</span>-cocycle <span class="SimpleMath">R_n ⟶ Z_q</span> where <span class="SimpleMath">G</span> acts trivially on <span class="SimpleMath">Z_q</span>. It is assumed <span class="SimpleMath">q=0</span> unless a value for <span class="SimpleMath">q</span> is entered. The command returns a function <span class="SimpleMath">F(g_1, ..., g_n)</span> which is the standard cocycle <span class="SimpleMath">G_n ⟶ Z_q</span> corresponding to <span class="SimpleMath">f</span>. At present the command is implemented only for <span class="SimpleMath">n=2</span> or <span class="SimpleMath">3</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Syzygy</code>( <var class="Arg">R</var>, <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R</span> (with contracting homotopy) and a list <span class="SimpleMath">g = [g[1], ..., g[n]]</span> of elements in <span class="SimpleMath">G</span>. It returns a word <span class="SimpleMath">w</span> in <spanclass="SimpleMath">R_n</span>. The word <span class="SimpleMath">w</span> is the image of the <spanclass="SimpleMath">n</span>-simplex in the standard bar resolution corresponding to the <span class="SimpleMath">n</span>-tuple <span class="SimpleMath">g</span>. This function can be used to construct explicit standard <span class="SimpleMath">n</span>-cocycles. (Currently implemented only for n<4.)</p>
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