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<div class="ChapSects"><a href="chap3.html#X792B9CC97C670AEA">3 <span class="Heading">Contracting Homotopies</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7E957D2381DBE362">3.1 <span class="Heading">The <code class="keyw">PartialContractingHomotopy</code> Data Type</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7D899ACD7EB512FA">3.1-1 ResolutionOfContractingHomotopy</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79C69C9B877C6D60">3.1-2 PartialContractingHomotopyLookup</a></span>
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<h3>3 <span class="Heading">Contracting Homotopies</span></h3>
<p><a id="X7E957D2381DBE362" name="X7E957D2381DBE362"></a></p>
<h4>3.1 <span class="Heading">The <code class="keyw">PartialContractingHomotopy</code> Data Type</span></h4>
<p>A partial contracting homotopy is a component object that knows the values of a contracting homotopy on some subspace of a resolution. It has two mandatory components:</p>
<ul>
<li><p><code class="code">.resolution</code> a <code class="keyw">HapResolution</code> on which the contraction is defined.</p>
</li>
<li><p><code class="code">.knownPartOfHomotopy</code> a list of <code class="keyw">Record</code>s with components <code class="code">.space</code> and <code class="code">.map</code>.</p>
</li>
</ul>
<p>Let <code class="code">h</code> be a contracting homotopy. The lookup table <code class="code">.knownPartOfHomotopy</code> has one entry for each term of the resolution <code class="code">h.resolution</code> (that is, one more than <code class="code">Length(h.resolution)</code>).</p>
<p>The <span class="SimpleMath">i</span> th element of <code class="code">.knownPartOfHomotopy</code> contains a record with components <code class="code">.space</code> and <code class="code">.map</code> where <code class="code">.space</code> is a <code class="keyw">FreeZGWord</code> of the <span class="SimpleMath">i-1</span> st term of the resolution. The component <code class="code">.map</code> is a list of length <code class="code">Dimension(h.resolution)(i-1)</code>. The entries of this list are pairs <code class="code">[g,im]</code> where <code class="code">g</code> represents a group element and <code class="code">im</code> represents the image of the contraction. So the entry <code class="code">[g,im]</code> in the <code class="code">k</code>th component of the list <code class="code">.map</code> means that the <code class="code">k</code>th free generator of the corresponding module multiplied with the group element represented by <code class="code">g</code> is mapped to <code class="code">im</code> under the partial contracting homotopy. Note that the data type of <code class="code">g</code> or <code class="code">im</code> are not fixed at this level. They must be specified by the sub representations. Also, <code class="code">im</code> need not represent the actual image under a contracting homotopy. It is possible to just store a bit of information that is then used to generate the actual image.</p>
<p>As this is a very general data type, it has very few methods.</p>
<p><a id="X7D899ACD7EB512FA" name="X7D899ACD7EB512FA"></a></p>
<h5>3.1-1 ResolutionOfContractingHomotopy</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionOfContractingHomotopy</code>( <var class="Arg">homotopy</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: A <code class="keyw">HapResolution</code></p>
<p>This returns the resolution of the homotopy <var class="Arg">homotopy</var> (the component <var class="Arg">homotopy!.resolution</var>).</p>
<p><a id="X79C69C9B877C6D60" name="X79C69C9B877C6D60"></a></p>
<h5>3.1-2 PartialContractingHomotopyLookup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PartialContractingHomotopyLookup</code>( <var class="Arg">homotopy</var>, <var class="Arg">term</var>, <var class="Arg">generator</var>, <var class="Arg">groupel</var> )</td><td class="tdright">( method )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PartialContractingHomotopyLookupNC</code>( <var class="Arg">homotopy</var>, <var class="Arg">term</var>, <var class="Arg">generator</var>, <var class="Arg">groupel</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns: The entry <code class="code">im</code> of the corresponding lookup table</p>
<p>Looks up the known part of the contracting homotopy <var class="Arg">homotopy</var> and returns the corresponding image. More precisely, it returns the image of the <var class="Arg">generator</var>th generator times the group element represented by <var class="Arg">groupel</var> in term <var class="Arg">term</var> under the partial homotopy. The data type of this image depends on the representation of <var class="Arg">homotopy</var>.</p>
<p><var class="Arg">term</var> has to be an integer and <var class="Arg">generator</var> a positive integer. <var class="Arg">groupel</var> only has to be an <code class="keyw">Object</code>.</p>
<p>The NC version does not do any checks on the input. The other version checks if <var class="Arg">term</var> and <var class="Arg">generator</var> are sensible. It does not check <var class="Arg">groupel</var>.</p>
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