Quelle  chap12.html   Sprache: HTML

<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (XMod) - Chapter 12: Applications</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap12"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap11.html">[Previous Chapter]</a>    <a href="chap13.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap12_mj.html">[MathJax on]</a></p>
<p><a id="X7DD7F0847FF2B96C" name="X7DD7F0847FF2B96C"></a></p>
<div class="ChapSects"><a href="chap12.html#X7DD7F0847FF2B96C">12 <span class="Heading">Applications</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X8575260A80F735BD">12.1 <span class="Heading">Free Loop Spaces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X87781C76804E783E">12.1-1 LoopClasses</a></span>
</div></div>
</div>

<h3>12 <span class="Heading">Applications</span></h3>

<p>This chapter was added in April 2018 for version 2.66 of <strong class="pkg">XMod</strong>. Initially it describes crossed modules for free loop spaces. Further applications may arise in due course.</p>

<p><a id="X8575260A80F735BD" name="X8575260A80F735BD"></a></p>

<h4>12.1 <span class="Heading">Free Loop Spaces</span></h4>

<p>These functions have been used to produce examples for Ronald Brown's paper <em>Crossed modules, and the homotopy <span class="SimpleMath">2</span>-type of a free loop space</em> <a href="chapBib.html#biBB18">[Bro18]</a>. The relevant theorem in that paper is as follows.</p>

<p><strong class="button">Theorem 2.1</strong> <em> Let <span class="SimpleMath">calM = (∂ : M -> P)</span> be a crossed module of groups and let <span class="SimpleMath">X = BcalM</span> be the classifying space of <span class="SimpleMath">calM</span>. Then the components of <span class="SimpleMath">LX</span>, the free loop space on <span class="SimpleMath">X</span>, are determined by equivalence classes of elements <span class="SimpleMath">a ∈ P</span> where <span class="SimpleMath">a,a'</span> are equivalent if and only if there are elements <span class="SimpleMath">m ∈ M, p ∈ P</span> such that <span class="SimpleMath">a'= p + a - ∂ m - p</span>. </em></p>

<p><em> Further the homotopy <span class="SimpleMath">2</span>-type of a component of <span class="SimpleMath">LX</span> given by <span class="SimpleMath">a ∈ P</span> is determined by the crossed module of groups <span class="SimpleMath">LcalM[a] = (∂_a : M -> P(a))</span> where: </em></p>


<ul>
<li><p><em> <span class="SimpleMath">P(a)</span> is the subgroup of the cat<span class="SimpleMath">^1</span>-group <span class="SimpleMath">G = P ⋉ M</span> such that <span class="SimpleMath">∂ m = [p,a] = -p-a+p+a</span>; </em></p>

</li>
<li><p><em> <span class="SimpleMath">∂_a(m) = (∂ m, m^-1m^a)</span> for <span class="SimpleMath">m ∈ M</span>; </em></p>

</li>
<li><p><em> the action of <span class="SimpleMath">P(a)</span> on <span class="SimpleMath">M</span> is given by <span class="SimpleMath">n^(p,m) = n^p</span> for <span class="SimpleMath">n ∈ M, (p,m) ∈ P(a)</span>. </em></p>

</li>
</ul>
<p><em> In particular <span class="SimpleMath">π_1(LX,a)</span> is isomorphic to <span class="SimpleMath">mathrmcokernel(∂_a)</span>, and <span class="SimpleMath">π_2(LX,a) ≅ π_2(X,*)^bara}</span>, the elements of <span class="SimpleMath">π_2(X,*)</span> fixed under the action of <span class="SimpleMath">bara</span>, the class of <span class="SimpleMath">a</span> in <span class="SimpleMath">π_1(X,*)</span>. </em></p>

<p><em> There is an exact sequence <span class="SimpleMath">π stackrelϕ-> π -> π_1(LX,a) -> C_bara}(π_1(X,*)) -> 1</span>, in which <span class="SimpleMath">π = π_2(X,*)</span>, and <span class="SimpleMath">ϕ</span> is the morphism <span class="SimpleMath">m ↦ m^-1m^a</span>. </em></p>

<p><a id="X87781C76804E783E" name="X87781C76804E783E"></a></p>

<h5>12.1-1 LoopClasses</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; LoopClasses</code>( <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; LoopsXMod</code>( <var class="Arg">M</var>, <var class="Arg">a</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllLoopsXMod</code>( <var class="Arg">M</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The operation <code class="code">LoopClasses</code> computes the equivalence classes <span class="SimpleMath">[a]</span> described above. These are all unions of conjugacy classes.</p>

<p>The operation <code class="code">LoopsXMod(M,a)</code> calculates the crossed module <span class="SimpleMath">LcalM[a]</span> described in the theorem.</p>

<p>The operation <code class="code">AllLoopsXMod(M)</code> returns a list of crossed modules, one for each equivalence class of elements <span class="SimpleMath">[a] ⊆ P</span>.</p>

<p>In the example below the automorphism crossed module <code class="code">X8</code> has <span class="SimpleMath">M ≅ C_2^3</span> and <span class="SimpleMath">P = PSL(3,2)</span> is the automorphism group of <span class="SimpleMath">M</span>. There are <span class="SimpleMath">6</span> equivalence classes which, in this case, are identical with the conjugacy classes. For each <span class="SimpleMath">LX</span> calculated, the <code class="func">IdGroup</code> (<a href="chap2.html#X7831DB527CF9DD57"><span class="RefLink">2.8-1</span></a>) is printed out.</p>


<div class="example"><pre>

<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( k8, "k8" ); </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Y8 := XModByAutomorphismGroup( k8 );; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">X8 := Image( IsomorphismPerm2DimensionalGroup( Y8 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetName( X8, "X8" );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Print( "X8: ", Size( X8 ), " : ", StructureDescription( X8 ), "\n" );  </span>
X8: [ 8, 168 ] : [ "C2 x C2 x C2", "PSL(3,2)" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">classes := LoopClasses( X8 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( classes, c -> Length(c) );</span>
[ 1, 21, 56, 42, 24, 24 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">LX := LoopsXMod( X8, (1,2)(5,6) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size2d( LX ); </span>
[ 8, 64 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( LX );</span>
[ [ 8, 5 ], [ 64, 138 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SetInfoLevel( InfoXMod, 1 );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LX8 := AllLoopsXMod( X8 );;</span>
#I  LoopsXMod with a = (),  IdGroup = [ [ 8, 5 ], [ 1344, 11686 ] ]
#I  LoopsXMod with a = (4,5)(6,7),  IdGroup = [ [ 8, 5 ], [ 64, 138 ] ]
#I  LoopsXMod with a = (2,3)(4,6,5,7),  IdGroup = [ [ 8, 5 ], [ 32, 6 ] ]
#I  LoopsXMod with a = (2,4,6)(3,5,7),  IdGroup = [ [ 8, 5 ], [ 24, 13 ] ]
#I  LoopsXMod with a = (1,2,4,3,6,7,5),  IdGroup = [ [ 8, 5 ], [ 56, 11 ] ]
#I  LoopsXMod with a = (1,2,4,5,7,3,6),  IdGroup = [ [ 8, 5 ], [ 56, 11 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">iso := IsomorphismGroups( Range( LX ), Range( LX8[2] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">iso = fail;</span>
false

</pre></div>


<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a>   <a href="chap0.html#contents">[Contents]</a>    <a href="chap11.html">[Previous Chapter]</a>    <a href="chap13.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chap6.html">6</a>  <a href="chap7.html">7</a>  <a href="chap8.html">8</a>  <a href="chap9.html">9</a>  <a href="chap10.html">10</a>  <a href="chap11.html">11</a>  <a href="chap12.html">12</a>  <a href="chap13.html">13</a>  <a href="chap14.html">14</a>  <a href="chap15.html">15</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>