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<div class="ChapSects"><a href="chap5_mj.html#X8735FC5E7BB5CE3A">5 <span class="Heading">Resolutions of the ground ring</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7CFDEEC07F15CF82">5.1 <span class="Heading"> </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7D8875C87BC9C379">5.1-1 TietzeReducedResolution</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X808535C3851CA4D4">5.1-2 ResolutionArithmeticGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79EA11238403019D">5.1-3 FreeGResolution</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8108E1047C31A058">5.1-4 ResolutionGTree</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7E4556B078B209CE">5.1-5 ResolutionSL2Z</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79CB82D77A1FAE9D">5.1-6 ResolutionAbelianGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79107B5F857DC27B">5.1-7 ResolutionAlmostCrystalGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X839D1B3B78B672BB">5.1-8 ResolutionAlmostCrystalQuotient</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X82A0D2B986724BB1">5.1-9 ResolutionArtinGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X87DABAF98575DC13">5.1-10 ResolutionAsphericalPresentation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7CA87AA478007468">5.1-11 ResolutionBieberbachGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7A20180E7D45038F">5.1-12 ResolutionCoxeterGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7FCE801781AD83E1">5.1-13 ResolutionDirectProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79C83C4881B6A656">5.1-14 ResolutionExtension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X87C346747F3B7C8C">5.1-15 ResolutionFiniteDirectProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79EDB1D584C2776F">5.1-16 ResolutionFiniteExtension</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X789B3E7C7CBB3751">5.1-17 ResolutionFiniteGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7AF48FA77F677E75">5.1-18 ResolutionFiniteSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X78E504557FD75664">5.1-19 ResolutionGraphOfGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7CBE6BDA7DB5AD7D">5.1-20 ResolutionNilpotentGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8574D76D7C891A04">5.1-21 ResolutionNormalSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X86934BE9858F7199">5.1-22 ResolutionPrimePowerGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X8521359A87D46462">5.1-23 ResolutionSmallFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X79A0221B7E96B642">5.1-24 ResolutionSubgroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X80CE971A7C2C538B">5.1-25 ResolutionSubnormalSeries</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X7A8E03D57895D04A">5.1-26 TwistedTensorProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X864044D679AE4E25">5.1-27 ConjugatedResolution</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5_mj.html#X82646B64875E5560">5.1-28 RecalculateIncidenceNumbers</a></span>
</div></div>
</div>
<h3>5 <span class="Heading">Resolutions of the ground ring</span></h3>
<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>
<h4>5.1 <span class="Heading"> </span></h4>
<p><a id="X7D8875C87BC9C379" name="X7D8875C87BC9C379"></a></p>
<h5>5.1-1 TietzeReducedResolution</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TietzeReducedResolution</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> and returns a <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution <span class="SimpleMath">\(S\)</span> which is obtained from <span class="SimpleMath">\(R\)</span> by applying "Tietze like operations" in each dimension. The hope is that <span class="SimpleMath">\(S\)</span> has fewer free generators than <span class="SimpleMath">\(R\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap13.html">1</a></span> </p>
<p><a id="X808535C3851CA4D4" name="X808535C3851CA4D4"></a></p>
<h5>5.1-2 ResolutionArithmeticGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionArithmeticGroup</code>( <var class="Arg">P</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a positive integer <span class="SimpleMath">\(n\)</span> and a string <span class="SimpleMath">\(P\)</span> equal to one of the following: <br /> <br /> "SL(2,Z)" , "SL(3,Z)" , "PGL(3,Z[i])" , "PGL(3,Eisenstein_Integers)" , "PSL(4,Z)" , "PSL(4,Z)_b" , "PSL(4,Z)_c" , "PSL(4,Z)_d" , "Sp(4,Z)" <br /> <br /> or the string <br /> <br /> "GL(2,O(-d))" <br /> <br /> for d=1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 43 <br /> <br /> or the string <br /> <br /> "SL(2,O(-d))" <br /> <br /> for d=2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 43, 67, 163 <br /> <br /> or the string <br /> <br /> "SL(2,O(-d))_a" <br /> <br /> for d=2, 7, 11, 19. <br /> <br /> It returns <span class="SimpleMath">\(n\)</span> terms of a free ZG-resolution for the group <span class="SimpleMath">\(G\)</span> described by the string. Here O(-d) denotes the ring of integers of Q(sqrt(-d)) and subscripts _a, _b , _c , _d denote alternative non-free ZG-resolutions for a given group G.<br /> <br /> Data for the first list of resolutions was provided provided by <strong class="button">Mathieu Dutour</strong>. Data for GL(2,O(-d)) was provided by <strong class="button">Sebastian Schoenennbeck</strong>. Data for SL(2,O(-d)) was provided by<strong class="button">Sebastian Schoennenbeck</strong> for d <= 26 and by <strong class="button">Alexander Rahm</strong> for d>26 and for the alternative complexes.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap13.html">1</a></span> </p>
<p><a id="X79EA11238403019D" name="X79EA11238403019D"></a></p>
<h5>5.1-3 FreeGResolution</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGResolution</code>( <var class="Arg">P</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">‣ FreeGResolution</code>( <var class="Arg">P</var>, <var class="Arg">n</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a non-free <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(P\)</span> with finite stabilizer groups, and a positive integer <span class="SimpleMath">\(n\)</span>. It returns a free <span class="SimpleMath">\(ZG\)</span>-resolution of length equal to the minimum of n and the length of <span class="SimpleMath">\(P\)</span>. If one requires only a mod <span class="SimpleMath">\(p\)</span> resolution then the prime <span class="SimpleMath">\(p\)</span> can be entered as an optional third argument.</p>
<p>The free resolution is returned without a contracting homotopy.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap1.html">1</a></span> , <span class="URL"><a href="../tutorial/chap6.html">2</a></span> , <span class="URL"><a href="../tutorial/chap7.html">3</a></span> , <span class="URL"><a href="../tutorial/chap11.html">4</a></span> , <span class="URL"><a href="../tutorial/chap13.html">5</a></span> , <span class="URL"><a href="../tutorial/chap14.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPolytopes.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutDavisComplex.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">10</a></span> </p>
<p><a id="X8108E1047C31A058" name="X8108E1047C31A058"></a></p>
<h5>5.1-4 ResolutionGTree</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionGTree</code>( <var class="Arg">P</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a non-free <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(P\)</span> of dimension 1 (i.e. a G-tree) with finite stabilizer groups, and a positive integer <span class="SimpleMath">\(n\)</span>. It returns a free <span class="SimpleMath">\(ZG\)</span>-resolution of length equal to n.</p>
<p>If <span class="SimpleMath">\(P\)</span> has a contracting homotopy then the free resolution is returned with a contracting homotopy.</p>
<p>This function was written by <strong class="button"> Bui Anh Tuan</strong>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X7E4556B078B209CE" name="X7E4556B078B209CE"></a></p>
<h5>5.1-5 ResolutionSL2Z</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSL2Z</code>( <var class="Arg">p</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs positive integers <span class="SimpleMath">\(m, n\)</span> and returns <span class="SimpleMath">\(n\)</span> terms of a <span class="SimpleMath">\(ZG\)</span>-resolution for the group <span class="SimpleMath">\(G=SL(2,Z[1/m])\)</span> .</p>
<p>This function is joint work with <strong class="button">Bui Anh Tuan</strong>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../tutorial/chap13.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutIntro.html">4</a></span> </p>
<p><a id="X79CB82D77A1FAE9D" name="X79CB82D77A1FAE9D"></a></p>
<h5>5.1-6 ResolutionAbelianGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionAbelianGroup</code>( <var class="Arg">L</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">‣ ResolutionAbelianGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a list <span class="SimpleMath">\(L:=[m_1,m_2, ..., m_d]\)</span> of nonnegative integers, and a positive integer <span class="SimpleMath">\(n\)</span>. It returns <span class="SimpleMath">\(n\)</span> terms of a <span class="SimpleMath">\({\mathbb Z}G\)</span>-resolution for the abelian group <span class="SimpleMath">\(G=Z_{L[1]}+Z_{L[2]}+···+{Z_L[d]}\)</span> .</p>
<p>If <span class="SimpleMath">\(G\)</span> is finite then the first argument can also be the abelian group <span class="SimpleMath">\(G\)</span> itself.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">3</a></span> </p>
<p><a id="X79107B5F857DC27B" name="X79107B5F857DC27B"></a></p>
<h5>5.1-7 ResolutionAlmostCrystalGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionAlmostCrystalGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a positive integer <span class="SimpleMath">\(n\)</span> and an almost crystallographic pcp group <span class="SimpleMath">\(G\)</span>. It returns <span class="SimpleMath">\(n\)</span> terms of a free <span class="SimpleMath">\(ZG\)</span>-resolution. (A group is almost crystallographic if it is nilpotent-by-finite and has no non-trivial finite normal subgroup. Such groups can be constructed using the ACLIB package.)</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">2</a></span> </p>
<p><a id="X839D1B3B78B672BB" name="X839D1B3B78B672BB"></a></p>
<h5>5.1-8 ResolutionAlmostCrystalQuotient</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionAlmostCrystalQuotient</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">c</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">‣ ResolutionAlmostCrystalQuotient</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">c</var>, <var class="Arg">false</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>An almost crystallographic group <span class="SimpleMath">\(G\)</span> is an extension of a finite group <span class="SimpleMath">\(P\)</span> by a nilpotent group <span class="SimpleMath">\(T\)</span>, and has no non-trivial finite normal subgroup. We define the relative lower central series by setting <span class="SimpleMath">\(T_1=T\)</span> and <span class="SimpleMath">\(T_{i+1}=[T_i,G]\)</span>.</p>
<p>This function inputs an almost crystallographic group <span class="SimpleMath">\(G\)</span> together with positive integers <span class="SimpleMath">\(n\)</span> and <span class="SimpleMath">\(c\)</span>. It returns <span class="SimpleMath">\(n\)</span> terms of a free <span class="SimpleMath">\(ZQ\)</span>-resolution <span class="SimpleMath">\(R\)</span> for the group <span class="SimpleMath">\(Q=G/T_c\)</span> .</p>
<p>In addition to the usual components, the resolution <span class="SimpleMath">\(R\)</span> has the component <span class="SimpleMath">\(R.quotientHomomorphism\)</span> which gives the quotient homomorphism <span class="SimpleMath">\(G \longrightarrow Q \)</span>.</p>
<p>If a fourth optional variable is set equal to "false" then the function omits to test whether <span class="SimpleMath">\(Q\)</span> is finite and a "more canonical" resolution is constructed.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">1</a></span> </p>
<p><a id="X82A0D2B986724BB1" name="X82A0D2B986724BB1"></a></p>
<h5>5.1-9 ResolutionArtinGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionArtinGroup</code>( <var class="Arg">D</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">\(D\)</span> and an integer <span class="SimpleMath">\(n>1\)</span>. It returns <span class="SimpleMath">\(n\)</span> terms of a free <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> where <span class="SimpleMath">\(G\)</span> is the Artin monoid associated to <span class="SimpleMath">\(D\)</span>. It is conjectured that <span class="SimpleMath">\(R\)</span> is also a free resolution for the Artin group <span class="SimpleMath">\(G\)</span>. The conjecture is known to hold in <span class="URL"><a href="../www/SideLinks/About/aboutArtinGroups.html">certain cases</a></span>.</p>
<p><span class="SimpleMath">\(G=R.group\)</span> is infinite and returned as a finitely presented group. The list <span class="SimpleMath">\(R.elts\)</span> is a partial listing of the elements of <span class="SimpleMath">\(G\)</span> which grows as <span class="SimpleMath">\(R\)</span> is used. Initially <span class="SimpleMath">\(R.elts\)</span> is empty and then, any time the boundary of a resolution generator is called, <span class="SimpleMath">\(R.elts\)</span> is updated to include elements of <span class="SimpleMath">\(G\)</span> involved in the boundary.</p>
<p>The contracting homotopy on <span class="SimpleMath">\(R\)</span> has not yet been implemented! Furthermore, the group <span class="SimpleMath">\(G\)</span> is currently returned only as a finitely presented group (without any method for solving the word problem).</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArtinGroups.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCohomologyRings.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">5</a></span> </p>
<p><a id="X87DABAF98575DC13" name="X87DABAF98575DC13"></a></p>
<h5>5.1-10 ResolutionAsphericalPresentation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionAsphericalPresentation</code>( <var class="Arg">F</var>, <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free group <span class="SimpleMath">\(F\)</span>, a set <span class="SimpleMath">\(R\)</span> of words in <span class="SimpleMath">\(F\)</span> which constitute an aspherical presentation for a group <span class="SimpleMath">\(G\)</span>, and a positive integer <span class="SimpleMath">\(n\)</span>. (Asphericity can be a difficult property to verify. The function <span class="SimpleMath">\(IsAspherical(F,R)\)</span> could be of help.)</p>
<p>The function returns n terms of a free <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> which has generators in dimensions < 3 only. No contracting homotopy on <span class="SimpleMath">\(R\)</span> will be returned.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap3.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutAspherical.html">3</a></span> </p>
<p><a id="X7CA87AA478007468" name="X7CA87AA478007468"></a></p>
<h5>5.1-11 ResolutionBieberbachGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionBieberbachGroup</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">‣ ResolutionBieberbachGroup</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a torsion free crystallographic group <span class="SimpleMath">\(G\)</span>, also known as a Bieberbach group, represented using AffineCrystGroupOnRight as in the GAP package Cryst. It also optionally inputs a choice of vector <span class="SimpleMath">\(v\)</span> in the euclidean space <span class="SimpleMath">\(R^n\)</span> on which <span class="SimpleMath">\(G\)</span> acts freely. The function returns <span class="SimpleMath">\(n+1\)</span> terms of the free <span class="SimpleMath">\(ZG\)</span>-resolution of <span class="SimpleMath">\(Z\)</span> arising as the cellular chain complex of the tesselation of <span class="SimpleMath">\(R^n\)</span> by the Dirichlet-Voronoi fundamental domain determined by <span class="SimpleMath">\(v\)</span>.</p>
<p>This function is part of the HAPcryst package written by <strong class="button">Marc Roeder</strong> and thus requires the HAPcryst package to be loaded.</p>
<p>The function requires the use of Polymake software.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> </p>
<p><a id="X7A20180E7D45038F" name="X7A20180E7D45038F"></a></p>
<h5>5.1-12 ResolutionCoxeterGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionCoxeterGroup</code>( <var class="Arg">D</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Coxeter diagram <span class="SimpleMath">\(D\)</span> and an integer <span class="SimpleMath">\(n>1\)</span>. It returns <span class="SimpleMath">\(k\)</span> terms of a free <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> where <span class="SimpleMath">\(G\)</span> is the Coxeter group associated to <span class="SimpleMath">\(D\)</span>. Here <span class="SimpleMath">\(k\)</span> is the maximum of n and the number of vertices in the Coxeter diagram. At present the implementation is only for finite Coxeter groups and the group <span class="SimpleMath">\(G\)</span> is returned as a permutation group. The contracting homotopy on <span class="SimpleMath">\(R\)</span> has not yet been implemented!</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap11.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPolytopes.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoxeter.html">3</a></span> </p>
<p><a id="X7FCE801781AD83E1" name="X7FCE801781AD83E1"></a></p>
<h5>5.1-13 ResolutionDirectProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionDirectProduct</code>( <var class="Arg">R</var>, <var class="Arg">S</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">‣ ResolutionDirectProduct</code>( <var class="Arg">R</var>, <var class="Arg">S</var>, <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> and <span class="SimpleMath">\(ZH\)</span>-resolution <span class="SimpleMath">\(S\)</span>. It outputs a <span class="SimpleMath">\(ZD\)</span>-resolution for the direct product <span class="SimpleMath">\(D=G x H\)</span>.</p>
<p>If <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(H\)</span> lie in a common group <span class="SimpleMath">\(K\)</span>, and if they commute and have trivial intersection, then an optional third variable <span class="SimpleMath">\(str\)</span>="internal" can be used. This will force <span class="SimpleMath">\(D\)</span> to be the subgroup <span class="SimpleMath">\(GH\)</span> in <span class="SimpleMath">\(K\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">1</a></span> </p>
<p><a id="X79C83C4881B6A656" name="X79C83C4881B6A656"></a></p>
<h5>5.1-14 ResolutionExtension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionExtension</code>( <var class="Arg">g</var>, <var class="Arg">R</var>, <var class="Arg">S</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">‣ ResolutionExtension</code>( <var class="Arg">g</var>, <var class="Arg">R</var>, <var class="Arg">S</var>, <var class="Arg">str</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">‣ ResolutionExtension</code>( <var class="Arg">g</var>, <var class="Arg">R</var>, <var class="Arg">S</var>, <var class="Arg">str</var>, <var class="Arg">GmapE</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a surjective group homomorphism <span class="SimpleMath">\(g:E \longrightarrow G\)</span> with kernel <span class="SimpleMath">\(N\)</span>. It also inputs a <span class="SimpleMath">\(ZN\)</span>-resolution <span class="SimpleMath">\(R\)</span> and a <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(S\)</span>. It returns a <span class="SimpleMath">\(ZE\)</span>-resolution. The groups <span class="SimpleMath">\(E\)</span> and <span class="SimpleMath">\(G\)</span> can be infinite.</p>
<p>If an optional fourth argument <span class="SimpleMath">\(str\)</span>is set equal to "TestFiniteness" then the groups <span class="SimpleMath">\(N\)</span> and <span class="SimpleMath">\(G\)</span> will be tested to see if they are finite. If they are finite then some speed saving routines will be invoked. One can also set <span class="SimpleMath">\(str\)</span>="NoTest".</p>
<p>If the homomorphism <span class="SimpleMath">\(g\)</span> is such that the GAP function <span class="SimpleMath">\(PreImagesElement(g,x)\)</span> doesn't work, then a function \(GmapE()\) should be included as a fifth input. For any \(x\) in \(G\) this function should return an element \(GmapE(x)\) in \(E\) which gets mapped onto \(x\) by \(g\).
<p>The contracting homotopy on the <span class="SimpleMath">\(ZE\)</span>-resolution has not yet been fully implemented for infinite groups!</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutRosenbergerMonster.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">2</a></span> </p>
<p><a id="X87C346747F3B7C8C" name="X87C346747F3B7C8C"></a></p>
<h5>5.1-15 ResolutionFiniteDirectProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionFiniteDirectProduct</code>( <var class="Arg">R</var>, <var class="Arg">S</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">‣ ResolutionFiniteDirectProduct</code>( <var class="Arg">R</var>, <var class="Arg">S</var>, <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> and <span class="SimpleMath">\(ZH\)</span>-resolution <span class="SimpleMath">\(S\)</span> where <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(H\)</span> are finite groups. It outputs a <span class="SimpleMath">\(ZD\)</span>-resolution for the direct product <span class="SimpleMath">\(D=G×H\)</span>.</p>
<p>If <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(H\)</span> lie in a common group <span class="SimpleMath">\(K\)</span>, and if they commute and have trivial intersection, then an optional third variable <span class="SimpleMath">\(str\)</span>="internal" can be used. This will force <span class="SimpleMath">\(D\)</span> to be the subgroup <span class="SimpleMath">\(GH\)</span> in <span class="SimpleMath">\(K\)</span>.</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X79EDB1D584C2776F" name="X79EDB1D584C2776F"></a></p>
<h5>5.1-16 ResolutionFiniteExtension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionFiniteExtension</code>( <var class="Arg">gensE</var>, <var class="Arg">gensG</var>, <var class="Arg">R</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">‣ ResolutionFiniteExtension</code>( <var class="Arg">gensE</var>, <var class="Arg">gensG</var>, <var class="Arg">R</var>, <var class="Arg">n</var>, <var class="Arg">true</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">‣ ResolutionFiniteExtension</code>( <var class="Arg">gensE</var>, <var class="Arg">gensG</var>, <var class="Arg">R</var>, <var class="Arg">n</var>, <var class="Arg">false</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs: a set <span class="SimpleMath">\(gensE\)</span> of generators for a finite group <span class="SimpleMath">\(E\)</span>; a set <span class="SimpleMath">\(gensG\)</span> equal to the image of <span class="SimpleMath">\(gensE\)</span> in a quotient group <span class="SimpleMath">\(G\)</span> of <span class="SimpleMath">\(E\)</span>; a <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> up to dimension at least <span class="SimpleMath">\(n\)</span>; a positive integer <span class="SimpleMath">\(n\)</span>. It uses the <span class="SimpleMath">\(TwistedTensorProduct()\)</span> construction to return <span class="SimpleMath">\(n\)</span> terms of a <span class="SimpleMath">\(ZE\)</span>-resolution.</p>
<p>The function has an optional fourth argument which, when set equal to "true", invokes tietze reductions in the construction of a resolution for the kernel of <span class="SimpleMath">\(E \longrightarrow G\)</span>.</p>
<p>If a <span class="SimpleMath">\(ZN\)</span>-resolution <span class="SimpleMath">\(S\)</span> is available, where <span class="SimpleMath">\(N\)</span> is the kernel of the quotient <span class="SimpleMath">\(E \longrightarrow G\)</span>, then this can be incorporated into the computations using an optional fifth argument.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutParallel.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">2</a></span> </p>
<p><a id="X789B3E7C7CBB3751" name="X789B3E7C7CBB3751"></a></p>
<h5>5.1-17 ResolutionFiniteGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionFiniteGroup</code>( <var class="Arg">gens</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">‣ ResolutionFiniteGroup</code>( <var class="Arg">gens</var>, <var class="Arg">n</var>, <var class="Arg">true</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">‣ ResolutionFiniteGroup</code>( <var class="Arg">gens</var>, <var class="Arg">n</var>, <var class="Arg">false</var>, <var class="Arg">p</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">‣ ResolutionFiniteGroup</code>( <var class="Arg">gens</var>, <var class="Arg">n</var>, <var class="Arg">false</var>, <var class="Arg">0</var>, <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a set <span class="SimpleMath">\(gens\)</span> of generators for a finite group <span class="SimpleMath">\(G\)</span> and a positive integer <span class="SimpleMath">\(n\)</span>. It outputs <span class="SimpleMath">\(n\)</span> terms of a <span class="SimpleMath">\(ZG\)</span>-resolution.</p>
<p>The function has an optional third argument which, when set equal to <span class="SimpleMath">\(true\)</span>, invokes tietze reductions in the construction of the resolution.</p>
<p>The function has an optional fourth argument which, when set equal to a prime <span class="SimpleMath">\(p\)</span>, records the fact that the resolution will only be used for mod <span class="SimpleMath">\(p\)</span> calculations. This could speed up subsequent constructions.</p>
<p>The function has an optional fifth argument <span class="SimpleMath">\(str\)</span> which, when set equal to "extendible", returns a resolution whose length can be increased using the command R!.extend() .</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap7.html">2</a></span> , <span class="URL"><a href="../tutorial/chap8.html">3</a></span> , <span class="URL"><a href="../tutorial/chap10.html">4</a></span> , <span class="URL"><a href="../tutorial/chap11.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutParallel.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPerformance.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCocycles.html">8</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPeriodic.html">9</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCohomologyRings.html">10</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPoincareSeries.html">11</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCrossedMods.html">12</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutDefinitions.html">13</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSimplicialGroups.html">14</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">15</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">16</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutFunctorial.html">17</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGouter.html">18</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTopology.html">19</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">20</a></span> </p>
<p><a id="X7AF48FA77F677E75" name="X7AF48FA77F677E75"></a></p>
<h5>5.1-18 ResolutionFiniteSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionFiniteSubgroup</code>( <var class="Arg">R</var>, <var class="Arg">K</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">‣ ResolutionFiniteSubgroup</code>( <var class="Arg">R</var>, <var class="Arg">gensG</var>, <var class="Arg">gensK</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(ZG\)</span>-resolution for a finite group <span class="SimpleMath">\(G\)</span> and a subgroup <span class="SimpleMath">\(K\)</span> of index <span class="SimpleMath">\(|G:K|\)</span>. It returns a free <span class="SimpleMath">\(ZK\)</span>-resolution whose <span class="SimpleMath">\(ZK\)</span>-rank is <span class="SimpleMath">\(|G:K|\)</span> times the <span class="SimpleMath">\(ZG\)</span>-rank in each dimension.</p>
<p>Generating sets <span class="SimpleMath">\(gensG\)</span>, <span class="SimpleMath">\(gensK\)</span> for <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(K\)</span> can also be input to the function (though the method does not depend on a choice of generators).</p>
<p>This <span class="SimpleMath">\(ZK\)</span>-resolution is not reduced. ie. it has more than one generator in dimension <span class="SimpleMath">\(0\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap8.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> , <span class="URL"><a href="../tutorial/chap13.html">3</a></span> , <span class="URL"><a href="../tutorial/chap14.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCoxeter.html">6</a></span> </p>
<p><a id="X78E504557FD75664" name="X78E504557FD75664"></a></p>
<h5>5.1-19 ResolutionGraphOfGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionGraphOfGroups</code>( <var class="Arg">D</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">‣ ResolutionGraphOfGroups</code>( <var class="Arg">D</var>, <var class="Arg">n</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a graph of groups <span class="SimpleMath">\(D\)</span> and a positive integer <span class="SimpleMath">\(n\)</span>. It returns <span class="SimpleMath">\(n\)</span> terms of a free <span class="SimpleMath">\(ZG\)</span>-resolution for the fundamental group <span class="SimpleMath">\(G\)</span> of <span class="SimpleMath">\(D\)</span>.</p>
<p>An optional third argument <span class="SimpleMath">\(L=[R_1 , \ldots , R_t]\)</span> can be used to list (in any order) free resolutions for some/all of the vertex and edge groups in <span class="SimpleMath">\(D\)</span>. If for some vertex or edge group no resolution is listed in <span class="SimpleMath">\(L\)</span> then the function <span class="SimpleMath">\(ResolutionFiniteGroup()\)</span> will be used to try to construct the resolution.</p>
<p>The <span class="SimpleMath">\(ZG\)</span>-resolution is usually not reduced. i.e. it has more than one generator in dimension 0.</p>
<p>The contracting homotopy on the <span class="SimpleMath">\(ZG\)</span>-resolution has not yet been implemented! Furthermore, the group <span class="SimpleMath">\(G\)</span> is currently returned only as a finitely presented group (without any method for solving the word problem).</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutGraphsOfGroups.html">3</a></span> </p>
<p><a id="X7CBE6BDA7DB5AD7D" name="X7CBE6BDA7DB5AD7D"></a></p>
<h5>5.1-20 ResolutionNilpotentGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionNilpotentGroup</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">‣ ResolutionNilpotentGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">str</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a nilpotent group <span class="SimpleMath">\(G\)</span> and positive integer <span class="SimpleMath">\(n\)</span>. It returns <span class="SimpleMath">\(n\)</span> terms of a free <span class="SimpleMath">\(ZG\)</span>-resolution. The resolution is computed using a divide-and-conquer technique involving the lower central series.</p>
<p>This function can be applied to infinite groups <span class="SimpleMath">\(G\)</span>. For finite groups the function <span class="SimpleMath">\(ResolutionNormalSeries()\)</span> probably gives better results.</p>
<p>If an optional third argument <span class="SimpleMath">\(str\)</span> is set equal to "TestFiniteness" then the groups <span class="SimpleMath">\(N\)</span> and <span class="SimpleMath">\(G\)</span> will be tested to see if they are finite. If they are finite then some speed saving routines will be invoked.</p>
<p>The contracting homotopy on the <span class="SimpleMath">\(ZE\)</span>-resolution has not yet been fully implemented for infinite groups.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutCohomologyRings.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRosenbergerMonster.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">5</a></span> </p>
<p><a id="X8574D76D7C891A04" name="X8574D76D7C891A04"></a></p>
<h5>5.1-21 ResolutionNormalSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionNormalSeries</code>( <var class="Arg">L</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">‣ ResolutionNormalSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</var>, <var class="Arg">true</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">‣ ResolutionNormalSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</var>, <var class="Arg">false</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a positive integer <span class="SimpleMath">\(n\)</span> and a list <span class="SimpleMath">\(L = [L_1 , ..., L_k]\)</span> of normal subgroups <span class="SimpleMath">\(L_i\)</span> of a finite group <span class="SimpleMath">\(G\)</span> satisfying <span class="SimpleMath">\(G = L_1\)</span> > <span class="SimpleMath">\(L2\)</span> ><span class="SimpleMath">\( \ldots \)</span> ><span class="SimpleMath">\( L_k\)</span>. Alternatively, <span class="SimpleMath">\(L = [gensL_1, ... gensL_k]\)</span> can be a list of generating sets for the <span class="SimpleMath">\(L_i\)</span> (and these particular generators will be used in the construction of resolutions). It returns a <span class="SimpleMath">\(ZG\)</span>-resolution by repeatedly using the function <span class="SimpleMath">\(ResolutionFiniteExtension()\)</span>.</p>
<p>The function has an optional third argument which, if set equal to true, invokes tietze reductions in the construction of resolutions.</p>
<p>The function has an optional fourth argument which, if set equal to p > 0, produces a resolution which is only valid for mod <span class="SimpleMath">\(p\)</span> calculations.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap10.html">2</a></span> , <span class="URL"><a href="../tutorial/chap11.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutModPRings.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPerformance.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutPersistent.html">6</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutRosenbergerMonster.html">7</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">8</a></span> </p>
<p><a id="X86934BE9858F7199" name="X86934BE9858F7199"></a></p>
<h5>5.1-22 ResolutionPrimePowerGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionPrimePowerGroup</code>( <var class="Arg">P</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">‣ ResolutionPrimePowerGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(P\)</span> and integer <span class="SimpleMath">\(n\)</span>><span class="SimpleMath">\(0\)</span>. It uses GAP's standard linear algebra functions over the field \(F\) of p elements to construct a free \(FP\)-resolution for mod \(p\) calculations only. The resolution is minimal - meaning that the number of generators of \(R_n\) equals the rank of \(H_n(P,F)\).
<p>The function can also be used to obtain a free non-minimal <span class="SimpleMath">\(FG\)</span>-resolution of a small nilpotent group <span class="SimpleMath">\(G\)</span> of non-prime-power order. In this case the prime <span class="SimpleMath">\(p\)</span> must be entered as the third input variable. (In the non-prime-power nilpotent case the algorithm is naive and not very good.)</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../tutorial/chap8.html">2</a></span> , <span class="URL"><a href="../tutorial/chap11.html">3</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutModPRings.html">4</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutDefinitions.html">5</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTopology.html">6</a></span> </p>
<p><a id="X8521359A87D46462" name="X8521359A87D46462"></a></p>
<h5>5.1-23 ResolutionSmallFpGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSmallFpGroup</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">‣ ResolutionSmallFpGroup</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a small finitely presented group <span class="SimpleMath">\(G\)</span> and an integer <span class="SimpleMath">\(n\)</span>><span class="SimpleMath">\(0\)</span>. It returns <span class="SimpleMath">\(n\)</span> terms of a <span class="SimpleMath">\(ZG\)</span>-resolution which, in dimensions 1 and 2, corresponds to the given presentation for <span class="SimpleMath">\(G\)</span>. The method returns no contracting homotopy for the resolution.</p>
<p>The function has an optional fourth argument which, when set equal to a prime <span class="SimpleMath">\(p\)</span>, records the fact that the resolution will only be used for mod <span class="SimpleMath">\(p\)</span> calculations. This could speed up subsequent constructions.</p>
<p>This function was written by Irina Kholodna.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutPeriodic.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutSpaceGroup.html">2</a></span> </p>
<p><a id="X79A0221B7E96B642" name="X79A0221B7E96B642"></a></p>
<h5>5.1-24 ResolutionSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSubgroup</code>( <var class="Arg">R</var>, <var class="Arg">K</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(ZG\)</span>-resolution for an (infinite) group <span class="SimpleMath">\(G\)</span> and a subgroup <span class="SimpleMath">\(K\)</span> of finite index <span class="SimpleMath">\(|G:K|\)</span>. It returns a free <span class="SimpleMath">\(ZK\)</span>-resolution whose <span class="SimpleMath">\(ZK\)</span>-rank is <span class="SimpleMath">\(|G:K|\)</span> times the <span class="SimpleMath">\(ZG\)</span>-rank in each dimension.</p>
<p>If <span class="SimpleMath">\(G\)</span> is finite then the function <span class="SimpleMath">\(ResolutionFiniteSubgroup(R,G,K)\)</span> will probably work better. In particular, resolutions from this function probably won't work with the function \(EquivariantChainMap()\). This \(ZK\)-resolution is not reduced. i.e. it has more than one generator in dimension 0.
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutArithmetic.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutArtinGroups.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutTwistedCoefficients.html">3</a></span> </p>
<p><a id="X80CE971A7C2C538B" name="X80CE971A7C2C538B"></a></p>
<h5>5.1-25 ResolutionSubnormalSeries</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSubnormalSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a positive integer n and a list <span class="SimpleMath">\(L = [L_1 , \ldots , L_k]\)</span> of subgroups <span class="SimpleMath">\(L_i\)</span> of a finite group <span class="SimpleMath">\(G=L_1\)</span> such that <span class="SimpleMath">\(L_1\)</span> > <span class="SimpleMath">\(L2 \ldots \)</span> > <span class="SimpleMath">\(L_k\)</span> is a subnormal series in <span class="SimpleMath">\(G\)</span> (meaning that each <span class="SimpleMath">\(L_{i+1}\)</span> must be normal in <span class="SimpleMath">\(L_i\)</span>). It returns a <span class="SimpleMath">\(ZG\)</span>-resolution by repeatedly using the function <span class="SimpleMath">\(ResolutionFiniteExtension()\)</span>.</p>
<p>If <span class="SimpleMath">\(L\)</span> is a series of normal subgroups in <span class="SimpleMath">\(G\)</span> then the function <span class="SimpleMath">\(ResolutionNormalSeries(L,n)\)</span> will possibly work more efficiently.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap6.html">1</a></span> , <span class="URL"><a href="../tutorial/chap11.html">2</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">3</a></span> </p>
<p><a id="X7A8E03D57895D04A" name="X7A8E03D57895D04A"></a></p>
<h5>5.1-26 TwistedTensorProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TwistedTensorProduct</code>( <var class="Arg">R</var>, <var class="Arg">S</var>, <var class="Arg">EhomG</var>, <var class="Arg">GmapE</var>, <var class="Arg">NhomE</var>, <var class="Arg">NEhomN</var>, <var class="Arg">EltsE</var>, <var class="Arg">Mult</var>, <var class="Arg">InvE</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a <span class="SimpleMath">\(ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span>, a <span class="SimpleMath">\(ZN\)</span>-resolution <span class="SimpleMath">\(S\)</span>, and other data relating to a short exact sequence <span class="SimpleMath">\(1 \longrightarrow N \longrightarrow E \longrightarrow G \longrightarrow 1\)</span>. It uses a perturbation technique of CTC Wall to construct a <span class="SimpleMath">\(ZE\)</span>-resolution <span class="SimpleMath">\(F\)</span>. Both <span class="SimpleMath">\(G\)</span> and <span class="SimpleMath">\(N\)</span> could be infinite. The "length" of <span class="SimpleMath">\(F\)</span> is equal to the minimum of the "length"s of <span class="SimpleMath">\(R\)</span> and <span class="SimpleMath">\(S\)</span>. The resolution <span class="SimpleMath">\(R\)</span> needs no contracting homotopy if no such homotopy is requied for <span class="SimpleMath">\(F\)</span>.</p>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../www/SideLinks/About/aboutExtensions.html">1</a></span> </p>
<p><a id="X864044D679AE4E25" name="X864044D679AE4E25"></a></p>
<h5>5.1-27 ConjugatedResolution</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ConjugatedResolution</code>( <var class="Arg">R</var>, <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a ZG-resoluton <span class="SimpleMath">\(R\)</span> and an element <span class="SimpleMath">\(x\)</span> from some group containing <span class="SimpleMath">\(G\)</span>. It returns a <span class="SimpleMath">\(ZG^x\)</span>-resolution <span class="SimpleMath">\(S\)</span> where the group <span class="SimpleMath">\(G^x\)</span> is the conjugate of <span class="SimpleMath">\(G\)</span> by <span class="SimpleMath">\(x\)</span>. (The component <span class="SimpleMath">\(S!.elts\)</span> will be a pseudolist rather than a list.)</p>
<p><strong class="button">Examples:</strong></p>
<p><a id="X82646B64875E5560" name="X82646B64875E5560"></a></p>
<h5>5.1-28 RecalculateIncidenceNumbers</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RecalculateIncidenceNumbers</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a ZG-resoluton <span class="SimpleMath">\(R\)</span> which arises as the cellular chain complex of a regular CW-complex. (Thus the boundary of any cell is a list of distinct cells.) It recalculates the incidence numbers for < | | |
