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<div class="ChapSects"><a href="chap14_mj.html#X78C726EB7F6CDAC0">14 <span class="Heading"> Cohomology rings of <span class="SimpleMath">\(p\)</span>-groups (mainly <span class="SimpleMath">\(p=2)\)</span></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14_mj.html#X7CFDEEC07F15CF82">14.1 <span class="Heading">  </span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14_mj.html#X831034A284F3906F">14.1-1 Mod2CohomologyRingPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14_mj.html#X7E1A4C8781A02CD0">14.1-2 PoincareSeriesLHS</a></span>
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<h3>14 <span class="Heading"> Cohomology rings of <span class="SimpleMath">\(p\)</span>-groups (mainly <span class="SimpleMath">\(p=2)\)</span></span></h3>

<p>The functions on this page were written by <strong class="button">Paul Smith</strong>. (They are included in HAP but they are also independently included in Paul Smiths HAPprime package.)</p>

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<h4>14.1 <span class="Heading">  </span></h4>

<p><a id="X831034A284F3906F" name="X831034A284F3906F"></a></p>

<h5>14.1-1 Mod2CohomologyRingPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">G</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">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">G</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">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">A</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">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>When applied to a finite <span class="SimpleMath">\(2\)</span>-group <span class="SimpleMath">\(G\)</span> this function returns a presentation for the mod 2 cohomology ring <span class="SimpleMath">\(H^*(G,Z_2)\)</span>. The Lyndon-Hochschild-Serre spectral sequence is used to prove that the presentation is correct.</p>

<p>When the function is applied to a <span class="SimpleMath">\(2\)</span>-group <span class="SimpleMath">\(G\)</span> and positive integer <span class="SimpleMath">\(n\)</span> the function first constructs <span class="SimpleMath">\(n\)</span> terms of a free <span class="SimpleMath">\(Z_2G\)</span>-resolution <span class="SimpleMath">\(R\)</span>, then constructs the finite-dimensional graded algebra <span class="SimpleMath">\(A=H^(*\le n)(G,Z_2)\)</span>, and finally uses <span class="SimpleMath">\(A\)</span> to approximate a presentation for <span class="SimpleMath">\(H^*(G,Z_2)\)</span>. For "sufficiently large" the approximation will be a correct presentation for <span class="SimpleMath">\(H^*(G,Z_2)\)</span>.</p>

<p>Alternatively, the function can be applied directly to either the resolution <span class="SimpleMath">\(R\)</span> or graded algebra <span class="SimpleMath">\(A\)</span>.</p>

<p>This function was written by <strong class="button">Paul Smith</strong>. It uses the Singular commutative algebra package to handle the Lyndon-Hochschild-Serre spectral sequence.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap8.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">2</a></span> </p>

<p><a id="X7E1A4C8781A02CD0" name="X7E1A4C8781A02CD0"></a></p>

<h5>14.1-2 PoincareSeriesLHS</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeriesLHS</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">\(2\)</span>-group <span class="SimpleMath">\(G\)</span> and returns a quotient of polynomials <span class="SimpleMath">\(f(x)=P(x)/Q(x)\)</span> whose coefficient of <span class="SimpleMath">\(x^k\)</span> equals the rank of the vector space <span class="SimpleMath">\(H_k(G,Z_2)\)</span> for all <span class="SimpleMath">\(k\)</span>.</p>

<p>This function was written by <strong class="button">Paul Smith</strong>. It use the Singular system for commutative algebra.</p>

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


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