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<div class="ChapSects"><a href="chap3_mj.html#X7F52C4747A402789">3 <span class="Heading">Basic functionality for homological group theory</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X85A9B66278AF63D9">3.1 <span class="Heading"> Cocycles</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X8343D6CA811C1E50">3.1-1 CcGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7C4C64EE864B04D5">3.1-2 CocycleCondition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7A69F5007F07F478">3.1-3 StandardCocycle</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7D02CE0A83211FB7">3.2 <span class="Heading"> G-Outer Groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X787C8FD6879771D9">3.2-1 ActedGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X8115386782214B38">3.2-2 ActingGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X847ABE6F781C7FE8">3.2-3 Centre</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X842035BD7E0B81EF">3.2-4 GOuterGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7C2A5A4F84DC70CB">3.3 <span class="Heading"> <span class="SimpleMath">\(G\)</span>-cocomplexes</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7D5E7FB97BF38DF1">3.3-1 CohomologyModule</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7CF7B8A3842D498B">3.3-2 HomToGModule</a></span>
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<h3>3 <span class="Heading">Basic functionality for homological group theory</span></h3>
<p>This page covers the functions used in chapter 4 of the book <span class="URL"><a href="https://global.oup.com/academic/product/an-invitation-to-computational-homotopy-9780198832980">An Invitation to Computational Homotopy</a></span>.</p>
<p><a id="X85A9B66278AF63D9" name="X85A9B66278AF63D9"></a></p>
<h4>3.1 <span class="Heading"> Cocycles</span></h4>
<p><a id="X8343D6CA811C1E50" name="X8343D6CA811C1E50"></a></p>
<h5>3.1-1 CcGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CcGroup</code>( <var class="Arg">N</var>, <var class="Arg">f</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(G\)</span>-outer group <span class="SimpleMath">\(N\)</span> with nonabelian cocycle describing some extension <span class="SimpleMath">\(N \rightarrowtail E \twoheadrightarrow G\)</span> together with standard 2-cocycle <span class="SimpleMath">\(f\colon G \times G \rightarrow A\)</span> where <span class="SimpleMath">\(A=Z(N)\)</span>. It returns the extension group determined by the cocycle <span class="SimpleMath">\(f\)</span>. The group is returned as a cocyclic group.</p>
<p>This function is part of the HAPcocyclic package of functions implemented by Robert F. Morse.</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>3.1-2 CocycleCondition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CocycleCondition</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> of <span class="SimpleMath">\(\mathbb Z\)</span> and an integer <span class="SimpleMath">\(n \ge 1\)</span>. It returns an integer matrix <span class="SimpleMath">\(M\)</span> with the following property. Let <span class="SimpleMath">\(d\)</span> be the <span class="SimpleMath">\(\mathbb ZG\)</span>-rank of <span class="SimpleMath">\(R_n\)</span>. An integer vector <span class="SimpleMath">\(f=[f_1, ... , f_d]\)</span> then represents a <span class="SimpleMath">\(\mathbb ZG\)</span>-homomorphism <span class="SimpleMath">\(R_n \rightarrow \mathbb Z_q\)</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">\(\mathbb ZG\)</span>-module <span class="SimpleMath">\(\mathbb Z_q=\mathbb Z/q{\mathbb Z}\)</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>3.1-3 StandardCocycle</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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 free <span class="SimpleMath">\(\mathbb ZG\)</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">\(R_n \rightarrow \mathbb Z_q=\mathbb Z/q\mathbb Z\)</span> where <span class="SimpleMath">\(G\)</span> acts trivially on <span class="SimpleMath">\(\mathbb Z_q\)</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">\(G^n \rightarrow \mathbb Z_q\)</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="X7D02CE0A83211FB7" name="X7D02CE0A83211FB7"></a></p>
<h4>3.2 <span class="Heading"> G-Outer Groups</span></h4>
<p><a id="X787C8FD6879771D9" name="X787C8FD6879771D9"></a></p>
<h5>3.2-1 ActedGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ActedGroup</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(G\)</span>-outer group <span class="SimpleMath">\(M\)</span> corresponding to a homomorphism <span class="SimpleMath">\(\alpha\colon G\rightarrow {\rm Out}(N)\)</span> and returns the group <span class="SimpleMath">\(N\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCrossedMods.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">4</a></span> </p>
<p><a id="X8115386782214B38" name="X8115386782214B38"></a></p>
<h5>3.2-2 ActingGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ActingGroup</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(G\)</span>-outer group <span class="SimpleMath">\(M\)</span> corresponding to a homomorphism <span class="SimpleMath">\(\alpha\colon G\rightarrow {\rm Out}(N)\)</span> and returns the group <span class="SimpleMath">\(G\)</span>.</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="X847ABE6F781C7FE8" name="X847ABE6F781C7FE8"></a></p>
<h5>3.2-3 Centre</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Centre</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(G\)</span>-outer group <span class="SimpleMath">\(M\)</span> and returns its group-theoretic centre as a <span class="SimpleMath">\(G\)</span>-outer group.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutParallel.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSchurMultiplier.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLieCovers.html">6</a></span> </p>
<p><a id="X842035BD7E0B81EF" name="X842035BD7E0B81EF"></a></p>
<h5>3.2-4 GOuterGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; GOuterGroup</code>( <var class="Arg">E</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">&#an style='color: green'>8227; GOuterGroup</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">\(E\)</span> and normal subgroup <span class="SimpleMath">\(N\)</span>. It returns <span class="SimpleMath">\(N\)</span> as a <span class="SimpleMath">\(G\)</span>-outer group where <span class="SimpleMath">\(G=E/N\)</span>. A nonabelian cocycle <span class="SimpleMath">\(f\colon G\times G\rightarrow N\)</span> is attached as a component of the <span class="SimpleMath">\(G\)</span>-Outer group.</p>
<p>The function can be used without an argument. In this case an empty outer group <span class="SimpleMath">\(C\)</span> is returned. The components must be set using <strong class="button">SetActingGroup(C,G)</strong>, <strong class="button">SetActedGroup(C,N)</strong> and <strong class="button">SetOuterAction(C,alpha)</strong>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoefficientSequence.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">4</a></span> </p>
<p><a id="X7C2A5A4F84DC70CB" name="X7C2A5A4F84DC70CB"></a></p>
<h4>3.3 <span class="Heading"> <span class="SimpleMath">\(G\)</span>-cocomplexes</span></h4>
<p><a id="X7D5E7FB97BF38DF1" name="X7D5E7FB97BF38DF1"></a></p>
<h5>3.3-1 CohomologyModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CohomologyModule</code>( <var class="Arg">C</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(G\)</span>-cocomplex <span class="SimpleMath">\(C\)</span> together with a non-negative integer <span class="SimpleMath">\(n\)</span>. It returns the cohomology <span class="SimpleMath">\(H^n(C)\)</span> as a <span class="SimpleMath">\(G\)</span>-outer group. If <span class="SimpleMath">\(C\)</span> was constructed from a <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> by homing to an abelian <span class="SimpleMath">\(G\)</span>-outer group <span class="SimpleMath">\(A\)</span> then, for each <span class="SimpleMath">\(x\)</span> in <span class="SimpleMath">\(H:=CohomologyModule(C,n)\)</span>, there is a function <span class="SimpleMath">\(f:=H!.representativeCocycle(x)\)</span> which is a standard <span class="SimpleMath">\(n\)</span>-cocycle corresponding to the cohomology class <span class="SimpleMath">\(x\)</span>. (At present this is implemented only for <span class="SimpleMath">\(n=1,2,3\)</span>.)</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/aboutCrossedMods.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">3</a></span> </p>
<p><a id="X7CF7B8A3842D498B" name="X7CF7B8A3842D498B"></a></p>
<h5>3.3-2 HomToGModule</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HomToGModule</code>( <var class="Arg">R</var>, <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> and an abelian <span class="SimpleMath">\(G\)</span>-outer group <span class="SimpleMath">\(A\)</span>. It returns the <span class="SimpleMath">\(G\)</span>-cocomplex obtained by applying <span class="SimpleMath">\(HomZG( \_ , A)\)</span>. (At present this function does not handle equivariant chain maps.)</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCrossedMods.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">4</a></span> </p>
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