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<?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 (HAP commands) - Chapter 9: Bredon homology</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="chap9" 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 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>9.2 <span class="Heading" >Arithmetic groups</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7DEBF2BB7D1FB144" >9.3 <span class="Heading" >Crystallographic groups</span ></a>span >div >div >span class="Heading" >Bredon homology</span ></h3>"X7B0212F97F3D442A" name="X7B0212F97F3D442A" ></a></p>span class="Heading" >Davis complex</span ></h4>span class="SimpleMath" >underline H_0(W,cal R) = Z^21</span ></p>span class="SimpleMath" >W</span > associated to the Dynkin diagram shown in the computation, with coefficients in the complex representation ring.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,6]]];;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >CoxeterDiagramDisplay(D);</span >pre ></div >img src="images/infcoxdiag.gif" align="center" height="160" alt="Coxeter diagram" /></p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >C:=DavisComplex(D);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >D:=TensorWithComplexRepresentationRing(C);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Homology(D,0);</span >pre ></div >"X7AFFB32587D047FE" name="X7AFFB32587D047FE" ></a></p>span class="Heading" >Arithmetic groups</span ></h4>span class="SimpleMath" >underline H_0(SL_2(cal O_-3),cal R) = Z_2⊕ Z^9</span ></p>span class="SimpleMath" >underline H_1(SL_2(cal O_-3),cal R) = Z</span ></p>span class="SimpleMath" >cal O_-3</span > the ring of integers of the number field <span class="SimpleMath" >Q(sqrt-3)</span >, and <span class="SimpleMath" >cal R</span > the complex reflection ring.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >R:=ContractibleGcomplex("SL(2,O-3)" );;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsRigid(R);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >S:=BaryCentricSubdivision(R);;</span span class="GAPprompt" >gap></span > <span class="GAPinput" >IsRigid(S);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >C:=TensorWithComplexRepresentationRing(S);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Homology(C,0);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Homology(C,1);</span >pre ></div >"X7DEBF2BB7D1FB144" name="X7DEBF2BB7D1FB144" ></a></p>span class="Heading" >Crystallographic groups</span ></h4>span class="SimpleMath" >underline H_0(G,cal R) = Z^17</span ></p>span class="SimpleMath" >G</span > the second crystallographic group of dimension <span class="SimpleMath" >4</span > in <strong class="button" >GAP</strong >'s library of crystallographic groups, and for cal R the Burnside ring. div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >G:=SpaceGroup(4,2);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >gens:=GeneratorsOfGroup(G);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >B:=CrystGFullBasis(G);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >R:=CrystGcomplex(gens,B,1);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsRigid(R);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >S:=CrystGcomplex(gens,B,0);;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >IsRigid(S);</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >D:=TensorWithBurnsideRing(S);;</span span class="GAPprompt" >gap></span > <span class="GAPinput" >Homology(D,0);</span >pre ></div >div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap8.html" >[Previous Chapter]</a> <a href="chap10.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a 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