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<p><a id="X7B54B8CA841C517B" name="X7B54B8CA841C517B"></a></p>
<div class="ChapSects"><a href="chap24.html#X7B54B8CA841C517B">24 <span class="Heading"> Cat-1-groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap24.html#X7CFDEEC07F15CF82">24.1 <span class="Heading">  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X79EB568C79A9EF01">24.1-1 AutomorphismGroupAsCatOneGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7F2E058F7AF17E82">24.1-2 HomotopyGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X866044CB7F43E1D2">24.1-3 HomotopyModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7FD749E97F92A32C">24.1-4 QuasiIsomorph</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X7DC403BA7DBA3ECE">24.1-5 ModuleAsCatOneGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X80CF81187D3A7C0A">24.1-6 MooreComplex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X857D6511876DEC0E">24.1-7 NormalSubgroupAsCatOneGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap24.html#X8213E0CA81209230">24.1-8 XmodToHAP</a></span>
</div></div>
</div>

<h3>24 <span class="Heading"> Cat-1-groups</span></h3>

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

<h4>24.1 <span class="Heading">  </span></h4>

<p><a id="X79EB568C79A9EF01" name="X79EB568C79A9EF01"></a></p>

<h5>24.1-1 AutomorphismGroupAsCatOneGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroupAsCatOneGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span> and returns the Cat-1-group <span class="SimpleMath">C</span> corresponding to the crossed module <span class="SimpleMath">G→ Aut(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="../tutorial/chap12.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/aboutquasi.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">6</a></span> </p>

<p><a id="X7F2E058F7AF17E82" name="X7F2E058F7AF17E82"></a></p>

<h5>24.1-2 HomotopyGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomotopyGroup</code>( <var class="Arg">C</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> and an integer n. It returns the <span class="SimpleMath">n</span>th homotopy group of <span class="SimpleMath">C</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/chap12.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutNonabelian.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCrossedMods.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutquasi.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTensorSquare.html">8</a></span> </p>

<p><a id="X866044CB7F43E1D2" name="X866044CB7F43E1D2"></a></p>

<h5>24.1-3 HomotopyModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomotopyModule</code>( <var class="Arg">C</var>, <var class="Arg">2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> and an integer n=2. It returns the second homotopy group of <span class="SimpleMath">C</span> as a G-module (i.e. abelian G-outer group) where G is the fundamental group of C.</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> </p>

<p><a id="X7FD749E97F92A32C" name="X7FD749E97F92A32C"></a></p>

<h5>24.1-4 QuasiIsomorph</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ QuasiIsomorph</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> and returns a cat-1-group <span class="SimpleMath">D</span> for which there exists some homomorphism <span class="SimpleMath">C→ D</span> that induces isomorphisms on homotopy groups.</p>

<p>This function was implemented by <strong class="button">Le Van Luyen</strong>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap12.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutquasi.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">3</a></span> </p>

<p><a id="X7DC403BA7DBA3ECE" name="X7DC403BA7DBA3ECE"></a></p>

<h5>24.1-5 ModuleAsCatOneGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModuleAsCatOneGroup</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span>, an abelian group <span class="SimpleMath">M</span> and a homomorphism <span class="SimpleMath">α: G→ Aut(M)</span>. It returns the Cat-1-group <span class="SimpleMath">C</span> corresponding th the zero crossed module <span class="SimpleMath">0: M→ G</span>.</p>

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

<p><a id="X80CF81187D3A7C0A" name="X80CF81187D3A7C0A"></a></p>

<h5>24.1-6 MooreComplex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MooreComplex</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> and returns its Moore complex as a G-complex (i.e. as a complex of groups considered as 1-outer groups).</p>

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

<p><a id="X857D6511876DEC0E" name="X857D6511876DEC0E"></a></p>

<h5>24.1-7 NormalSubgroupAsCatOneGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NormalSubgroupAsCatOneGroup</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">G</span> with normal subgroup <span class="SimpleMath">N</span>. It returns the Cat-1-group <span class="SimpleMath">C</span> corresponding th the inclusion crossed module <span class="SimpleMath">N→ G</span>.</p>

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

<p><a id="X8213E0CA81209230" name="X8213E0CA81209230"></a></p>

<h5>24.1-8 XmodToHAP</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XmodToHAP</code>( <var class="Arg">C</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a cat-1-group <span class="SimpleMath">C</span> obtained from the Xmod package and returns a cat-1-group <span class="SimpleMath">D</span> for which IsHapCatOneGroup(D) returns true.</p>

<p>It returns "fail" id <span class="SimpleMath">C</span> has not been produced by the Xmod package.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutquasi.html">1</a></span> </p>


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