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<div class="ChapSects"><a href="chap14.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.html#X7CFDEEC07F15CF82">14.1 <span class="Heading"> </span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.html#X831034A284F3906F">14.1-1 Mod2CohomologyRingPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap14.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>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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">&#an style='color: green'>8227; 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^(*≤ 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>
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<h5>14.1-2 PoincareSeriesLHS</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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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