1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
|
<?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 href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <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>
<p id="mathjaxlink" class="pcenter"><a href="chap9_mj.html">[MathJax on]</a></p>
<p><a id="X786DB80A8693779E" name="X786DB80A8693779E"></a></p>
<div class="ChapSects"><a href="chap9.html#X786DB80A8693779E">9 <span class="Heading">Bredon homology</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7B0212F97F3D442A">9.1 <span class="Heading">Davis complex</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X7AFFB32587D047FE">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>
<h3>9 <span class="Heading">Bredon homology</span></h3>
<p><a id="X7B0212F97F3D442A" name="X7B0212F97F3D442A"></a></p>
<h4>9.1 <span class="Heading">Davis complex</span></h4>
<p>The following example computes the Bredon homology</p>
<p><span class="SimpleMath">underline H_0(W,cal R) = Z^21</span></p>
<p>for the infinite Coxeter group <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>
<p><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>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
</pre></div>
<p><a id="X7AFFB32587D047FE" name="X7AFFB32587D047FE"></a></p>
<h4>9.2 <span class="Heading">Arithmetic groups</span></h4>
<p>The following example computes the Bredon homology</p>
<p><span class="SimpleMath">underline H_0(SL_2(cal O_-3),cal R) = Z_2⊕ Z^9</span></p>
<p><span class="SimpleMath">underline H_1(SL_2(cal O_-3),cal R) = Z</span></p>
<p>for <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>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=BaryCentricSubdivision(R);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRigid(S);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=TensorWithComplexRepresentationRing(S);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(C,0);</span>
[ 2, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(C,1);</span>
[ 0 ]
</pre></div>
<p><a id="X7DEBF2BB7D1FB144" name="X7DEBF2BB7D1FB144"></a></p>
<h4>9.3 <span class="Heading">Crystallographic groups</span></h4>
<p>The following example computes the Bredon homology</p>
<p><span class="SimpleMath">underline H_0(G,cal R) = Z^17</span></p>
<p>for <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 <span class="SimpleMath">cal R</span> the Burnside ring.</p>
<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>
false
<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>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=TensorWithBurnsideRing(S);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,0);</span>
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
</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 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 href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|
