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<div class="ChapSects"><a href="chap19_mj.html#X85A9B66278AF63D9">19 <span class="Heading"> Cocycles</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap19_mj.html#X7CFDEEC07F15CF82">19.1 <span class="Heading">  </span></a>
</span>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19_mj.html#X8343D6CA811C1E50">19.1-1 CcGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19_mj.html#X7C4C64EE864B04D5">19.1-2 CocycleCondition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19_mj.html#X7A69F5007F07F478">19.1-3 StandardCocycle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap19_mj.html#X8157A77284F56BAD">19.1-4 Syzygy</a></span>
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<h3>19 <span class="Heading"> Cocycles</span></h3>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>19.1 <span class="Heading">  </span></h4>

<p><a id="X8343D6CA811C1E50" name="X8343D6CA811C1E50"></a></p>

<h5>19.1-1 CcGroup</h5>

<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>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">2</a></span> </p>

<p><a id="X7C4C64EE864B04D5" name="X7C4C64EE864B04D5"></a></p>

<h5>19.1-2 CocycleCondition</h5>

<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">\(f=[f_1, \ldots , f_d]\)</span> then represents a <span class="SimpleMath">\(ZG\)</span>-homomorphism <span class="SimpleMath">\(R_n \longrightarrow Z_q\)</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>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCocycles.html">2</a></span> </p>

<p><a id="X7A69F5007F07F478" name="X7A69F5007F07F478"></a></p>

<h5>19.1-3 StandardCocycle</h5>

<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">\(R_n \longrightarrow Z_q\)</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">\(G_n \longrightarrow Z_q\)</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>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCocycles.html">2</a></span> </p>

<p><a id="X8157A77284F56BAD" name="X8157A77284F56BAD"></a></p>

<h5>19.1-4 Syzygy</h5>

<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 <span class="SimpleMath">\(R_n\)</span>. The word <span class="SimpleMath">\(w\)</span> is the image of the <span class="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>

<p><strong class="button">Examples:</strong></p>


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