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<div class="ChapSects"><a href="chap7.html#X787E37187B7308C9">7 <span class="Heading">Cohomology of groups (and Lie Algebras)</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X807B265978F90E01">7.1 <span class="Heading">Finite groups </span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X80A721AC7A8D30A3">7.1-1 <span class="Heading">Naive homology computation for a very small group</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X838CEA3F850DFC82">7.1-2 <span class="Heading">A more efficient homology computation</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X842E93467AD09EC1">7.1-3 <span class="Heading">Computation of an induced homology homomorphism</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X8754D2937E6FD7CE">7.1-4 <span class="Heading">Some other finite group homology computations</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X8463EF6A821FFB69">7.2 <span class="Heading">Nilpotent groups</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X82E8FAC67BC16C01">7.3 <span class="Heading">Crystallographic and Almost Crystallographic groups</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7AFFB32587D047FE">7.4 <span class="Heading">Arithmetic groups</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X800CB6257DC8FB3A">7.5 <span class="Heading">Artin groups</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7C3DEDD57BB4D537">7.8 <span class="Heading">Cohomology with coefficients in a module</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7E573EA582CCEF2E">7.9 <span class="Heading">Cohomology as a functor of the first variable</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X796731727A7EBE59">7.10 <span class="Heading">Cohomology as a functor of the second variable and the long exact coefficient sequence</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X80F6FD3E7C7E4E8D">7.11 <span class="Heading">Transfer Homomorphism</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X79B1406C803FF178">7.12 <span class="Heading">Cohomology rings of finite fundamental groups of 3-manifolds
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X833A19F0791C3B06">7.13 <span class="Heading">Explicit cocycles </span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7C5233E27D2D603E">7.14 <span class="Heading">Quillen's complex and the <span class="SimpleMath">p</span>-part of homology </span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X865CC8E0794C0E61">7.15 <span class="Heading">Homology of a Lie algebra</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X86B4EE4783A244F7">7.16 <span class="Heading">Covers of Lie algebras</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7DFF32A67FF39C82">7.16-1 <span class="Heading">Computing a cover</span></a>
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<h3>7 <span class="Heading">Cohomology of groups (and Lie Algebras)</span></h3>
<p><a id="X807B265978F90E01" name="X807B265978F90E01"></a></p>
<h4>7.1 <span class="Heading">Finite groups </span></h4>
<p><a id="X80A721AC7A8D30A3" name="X80A721AC7A8D30A3"></a></p>
<h5>7.1-1 <span class="Heading">Naive homology computation for a very small group</span></h5>
<p>It is possible to compute the low degree (co)homology of a finite group or monoid of small order directly from the bar resolution. The following commands take this approach to computing the fifth integral homology</p>
<p><span class="SimpleMath">H_5(Q_4, Z) = Z_2⊕ Z_2</span></p>
<p>of the quaternion group <span class="SimpleMath">G=Q_4</span> of order <span class="SimpleMath">8</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Q:=QuaternionGroup(8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:=BarComplexOfMonoid(Q,6);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=ContractedComplex(B);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(C,5);</span>
[ 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..6],B!.dimension);</span>
[ 1, 7, 49, 343, 2401, 16807, 117649 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..6],C!.dimension);</span>
[ 1, 2, 2, 1, 2, 4, 102945 ]
</pre></div>
<p>However, this approach is of limited applicability since the bar resolution involves <span class="SimpleMath">|G|^k</span> free generators in degree <span class="SimpleMath">k</span>. A range of techniques, tailored to specific classes of groups, can be used to compute the (co)homology of larger finite groups.</p>
<p>This naive approach does have the merit of being applicable to arbitrary small monoids. The following calculates the homology in degrees <span class="SimpleMath">≤ 7</span> of a monoid of order 8, the monoid being specified by its multiplication table.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:=[ [ 1, 1, 1, 4, 4, 4, 4, 1 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 1, 1, 4, 4, 4, 4, 2 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 1, 1, 4, 4, 4, 4, 3 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 4, 4, 4, 1, 1, 1, 1, 4 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 4, 4, 4, 1, 1, 1, 1, 5 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 4, 4, 4, 1, 1, 1, 1, 6 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 4, 4, 4, 1, 1, 1, 1, 7 ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 2, 3, 4, 5, 6, 7, 8 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=MonoidByMultiplicationTable(T);</span>
<monoid of size 8, with 8 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:=BarComplexOfMonoid(M,8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=ContractedComplex(B);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..7],i->Homology(C,i));</span>
[ [ 0 ], [ 2 ], [ ], [ 2 ], [ ], [ 2 ], [ ], [ 2 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..8],B!.dimension);</span>
[ 1, 7, 49, 343, 2401, 16807, 117649, 823543, 5764801 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..8],C!.dimension);</span>
[ 1, 1, 1, 1, 1, 1, 1, 1, 5044201 ]
</pre></div>
<p><a id="X838CEA3F850DFC82" name="X838CEA3F850DFC82"></a></p>
<h5>7.1-2 <span class="Heading">A more efficient homology computation</span></h5>
<p>The following example computes the seventh integral homology</p>
<p><span class="SimpleMath">H_7(M_23, Z) = Z_16⊕ Z_15</span></p>
<p>and fourth integral cohomomogy</p>
<p><span class="SimpleMath">H^4(M_24, Z) = Z_12</span></p>
<p>of the Mathieu groups <span class="SimpleMath">M_23</span> and <span class="SimpleMath">M_24</span>. (Warning: the computation of <span class="SimpleMath">H_7(M_23, Z)</span> takes a couple of hours to run.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupHomology(MathieuGroup(23),7);</span>
[ 16, 3, 5 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupCohomology(MathieuGroup(24),4);</span>
[ 4, 3 ]
</pre></div>
<p><a id="X842E93467AD09EC1" name="X842E93467AD09EC1"></a></p>
<h5>7.1-3 <span class="Heading">Computation of an induced homology homomorphism</span></h5>
<p>The following example computes the cokernel</p>
<p><span class="SimpleMath">coker( H_3(A_7, Z) → H_3(S_10, Z)) ≅ Z_2⊕ Z_2</span></p>
<p>of the degree-3 integral homomogy homomorphism induced by the canonical inclusion <span class="SimpleMath">A_7 → S_10</span> of the alternating group on <span class="SimpleMath">7</span> letters into the symmetric group on <span class="SimpleMath">10</span> letters. The analogous cokernel with <span class="SimpleMath">Z_2</span> homology coefficients is also computed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SymmetricGroup(10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=AlternatingGroup(7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=GroupHomomorphismByFunction(H,G,x->x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=GroupHomology(f,3);</span>
MappingByFunction( Pcp-group with orders [ 4, 3 ], Pcp-group with orders
[ 2, 2, 4, 3 ], function( x ) ... end )
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(Range(F)/Image(F));</span>
[ 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Fmod2:=GroupHomology(f,3,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(Range(Fmod2)/Image(Fmod2));</span>
[ 2, 2 ]
</pre></div>
<p><a id="X8754D2937E6FD7CE" name="X8754D2937E6FD7CE"></a></p>
<h5>7.1-4 <span class="Heading">Some other finite group homology computations</span></h5>
<p>The following example computes the third integral homology of the Weyl group <span class="SimpleMath">W=Weyl(E_8)</span>, a group of order <span class="SimpleMath">696729600</span>.</p>
<p><span class="SimpleMath">H_3(Weyl(E_8), Z) = Z_2 ⊕ Z_2 ⊕ Z_12</span></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=SimpleLieAlgebra("E",8,Rationals);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=WeylGroup(RootSystem(L));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(W);</span>
696729600
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupHomology(W,3);</span>
[ 2, 2, 4, 3 ]
</pre></div>
<p>The preceding calculation could be achieved more quickly by noting that <span class="SimpleMath">W=Weyl(E_8)</span> is a Coxeter group, and by using the associated Coxeter polytope. The following example uses this approach to compute the fourth integral homology of <span class="SimpleMath">W</span>. It begins by displaying the Coxeter diagram of <span class="SimpleMath">W</span>, and then computes</p>
<p><span class="SimpleMath">H_4(Weyl(E_8), Z) = Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_2</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=[[1,[2,3]],[2,[3,3]],[3,[4,3],[5,3]],[5,[6,3]],[6,[7,3]],[7,[8,3]]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CoxeterDiagramDisplay(D);</span>
</pre></div>
<p><img src="images/e8diagram.gif" align="center" height="200" alt="Coxeter diagram for E8"/></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">polytope:=CoxeterComplex_alt(D,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=FreeGResolution(polytope,5);</span>
Resolution of length 5 in characteristic 0 for <matrix group with
8 generators> .
No contracting homotopy available.
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=TensorWithIntegers(R);</span>
Chain complex of length 5 in characteristic 0 .
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(C,4);</span>
[ 2, 2, 2, 2 ]
</pre></div>
<p>The following example computes the sixth mod-<span class="SimpleMath">2</span> homology of the Sylow <span class="SimpleMath">2</span>-subgroup <span class="SimpleMath">Syl_2(M_24)</span> of the Mathieu group <span class="SimpleMath">M_24</span>.</p>
<p><span class="SimpleMath">H_6(Syl_2(M_24), Z_2) = Z_2^143</span></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupHomology(SylowSubgroup(MathieuGroup(24),2),6,2);</span>
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
</pre></div>
<p>The following example computes the sixth mod-<span class="SimpleMath">2</span> homology of the Unitary group <span class="SimpleMath">U_3(4)</span> of order 312000.</p>
<p><span class="SimpleMath">H_6(U_3(4), Z_2) = Z_2^4</span></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=GU(3,4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(G);</span>
312000
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupHomology(G,6,2);</span>
[ 2, 2, 2, 2 ]
</pre></div>
<p>The following example constructs the Poincare series</p>
<p><span class="SimpleMath">p(x)=frac1-x^3+3*x^2-3*x+1</span></p>
<p>for the cohomology <span class="SimpleMath">H^∗(Syl_2(M_12, F_2)</span>. The coefficient of <span class="SimpleMath">x^n</span> in the expansion of <span class="SimpleMath">p(x)</span> is equal to the dimension of the vector space <span class="SimpleMath">H^n(Syl_2(M_12, F_2)</span>. The computation involves <strong class="button">Singular</strong>'s Groebner basis algorithms and the Lyndon-Hochschild-Serre spectral sequence.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SylowSubgroup(MathieuGroup(12),2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P:=PoincareSeriesLHS(G);</span>
(1)/(-x_1^3+3*x_1^2-3*x_1+1)
</pre></div>
<p>The additional following command uses the Poincare series</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RankHomologyPGroup(G,P,1000);</span>
251000
</pre></div>
<p>to determine that <span class="SimpleMath">H_1000(Syl_2(M_12, Z)</span> is a direct sum of 251000 non-trivial cyclic <span class="SimpleMath">2</span>-groups.</p>
<p>The following example constructs the series</p>
<p><span class="SimpleMath">p(x)=fracx^4-x^3+x^2-x+1x^6-x^5+x^4-2*x^3+x^2-x+1</span></p>
<p>whose coefficient of <span class="SimpleMath">x^n</span> is equal to the dimension of the vector space <span class="SimpleMath">H^n(M_11, F_2)</span> for all <span class="SimpleMath">n</span> in the range <span class="SimpleMath">0≤ n≤ 14</span>. The coefficient is not guaranteed correct for <span class="SimpleMath">n≥ 15</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PoincareSeriesPrimePart(MathieuGroup(11),2,14);</span>
(x_1^4-x_1^3+x_1^2-x_1+1)/(x_1^6-x_1^5+x_1^4-2*x_1^3+x_1^2-x_1+1)
</pre></div>
<p><a id="X8463EF6A821FFB69" name="X8463EF6A821FFB69"></a></p>
<h4>7.2 <span class="Heading">Nilpotent groups</span></h4>
<p>The following example computes</p>
<p><span class="SimpleMath">H_4(N, Z) = (Z_3)^4 ⊕ Z^84</span></p>
<p>for the free nilpotent group <span class="SimpleMath">N</span> of class <span class="SimpleMath">2</span> on four generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup(4);; N:=NilpotentQuotient(F,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupHomology(N,4);</span>
[ 3, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 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="X82E8FAC67BC16C01" name="X82E8FAC67BC16C01"></a></p>
<h4>7.3 <span class="Heading">Crystallographic and Almost Crystallographic groups</span></h4>
<p>The following example computes</p>
<p><span class="SimpleMath">H_5(G, Z) = Z_2 ⊕ Z_2</span></p>
<p>for the <span class="SimpleMath">3</span>-dimensional crystallographic space group <span class="SimpleMath">G</span> with Hermann-Mauguin symbol "P62"</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupHomology(SpaceGroupBBNWZ("P62"),5);</span>
[ 2, 2 ]
</pre></div>
<p>The following example computes</p>
<p><span class="SimpleMath">H^5(G, Z)= Z</span></p>
<p>for an almost crystallographic group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=AlmostCrystallographicPcpGroup( 4, 50, [ 1, -4, 1, 2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GroupCohomology(G,4);</span>
[ 0 ]
</pre></div>
<p><a id="X7AFFB32587D047FE" name="X7AFFB32587D047FE"></a></p>
<h4>7.4 <span class="Heading">Arithmetic groups</span></h4>
<p>The following example computes</p>
<p><span class="SimpleMath">H_6(SL_2(cal O, Z) = Z_2 ⊕ Z_12</span></p>
<p>for <span class="SimpleMath">cal O</span> the ring of integers of the number field <span class="SimpleMath">Q(sqrt-2)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=ContractibleGcomplex("SL(2,O-2)");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=FreeGResolution(C,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(TensorWithIntegers(R),6);</span>
[ 2, 12 ]
</pre></div>
<p><a id="X800CB6257DC8FB3A" name="X800CB6257DC8FB3A"></a></p>
<h4>7.5 <span class="Heading">Artin groups</span></h4>
<p>The following example computes</p>
<p><span class="SimpleMath">H_n(G, Z) ={ beginarrayll Z &n=0,1,7,8 Z_2, &n=2,3 Z_2⊕ Z_6, &n=4,6 Z_3 ⊕ Z_6,& n=5 0, &n>8 endarray.</span></p>
<p>for <span class="SimpleMath">G</span> the Artin group of type <span class="SimpleMath">E_8</span>. (Similar commands can be used to compute a resolution and homology of arbitrary Artin monoids and, in thoses cases such as the spherical cases where the <span class="SimpleMath">K(π,1)</span>-conjecture is known to hold, the homology is equal to that of the corresponding Artin group.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=[[1,[2,3]],[2,[3,3]],[3,[4,3],[5,3]],[5,[6,3]],[6,[7,3]],[7,[8,3]]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">CoxeterDiagramDisplay(D);;</span>
</pre></div>
<p><img src="images/e8diagram.gif" align="center" height="200" alt="Coxeter diagram for E8"/></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionArtinGroup(D,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=TensorWithIntegers(R);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..8],n->Homology(C,n));</span>
[ [ 0 ], [ 0 ], [ 2 ], [ 2 ], [ 2, 6 ], [ 3, 6 ], [ 2, 6 ], [ 0 ], [ 0 ] ]
</pre></div>
<p>The Artin group <span class="SimpleMath">G</span> projects onto the Coxeter group <span class="SimpleMath">W</span> of type <span class="SimpleMath">E_8</span>. The group <span class="SimpleMath">W</span> has a natural representation as a group of <span class="SimpleMath">8× 8</span> integer matrices. This projection gives rise to a representation <span class="SimpleMath">ρ: G→ GL_8( Z)</span>. The following command computes the cohomology group <span class="SimpleMath">H^6(G,ρ) = ( Z_2)^6</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=R!.group;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gensG:=GeneratorsOfGroup(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">W:=CoxeterDiagramMatCoxeterGroup(D);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gensW:=GeneratorsOfGroup(W);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rho:=GroupHomomorphismByImages(G,W,gensG,gensW);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=HomToIntegralModule(R,rho);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology(C,6);</span>
[ 2, 2, 2, 2, 2, 2 ]
</pre></div>
<p><a id="X7BAFCA3680E478AE" name="X7BAFCA3680E478AE"></a></p>
<h4>7.6 <span class="Heading">Graphs of groups</span></h4>
<p>The following example computes</p>
<p><span class="SimpleMath">H_5(G, Z) = Z_2⊕ Z_2⊕ Z_2 ⊕ Z_2 ⊕ Z_2</span></p>
<p>for <span class="SimpleMath">G</span> the graph of groups corresponding to the amalgamated product <span class="SimpleMath">G=S_5*_S_3S_4</span> of the symmetric groups <span class="SimpleMath">S_5</span> and <span class="SimpleMath">S_4</span> over the canonical subgroup <span class="SimpleMath">S_3</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S5:=SymmetricGroup(5);SetName(S5,"S5");</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S4:=SymmetricGroup(4);SetName(S4,"S4");</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=SymmetricGroup(3);SetName(A,"S3");</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AS5:=GroupHomomorphismByFunction(A,S5,x->x);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">AS4:=GroupHomomorphismByFunction(A,S4,x->x);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=[S5,S4,[AS5,AS4]];</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">GraphOfGroupsDisplay(D);</span>
</pre></div>
<p><img src="images/graphgroups.png" align="center" height="100" alt="graph of groups"/></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionGraphOfGroups(D,6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(TensorWithIntegers(R),5);</span>
[ 2, 2, 2, 2, 2 ]
</pre></div>
<p><a id="X7CE849E58706796C" name="X7CE849E58706796C"></a></p>
<h4>7.7 <span class="Heading">Lie algebra homology and free nilpotent groups</span></h4>
<p>One method of producting a Lie algebra <span class="SimpleMath">L</span> from a group <span class="SimpleMath">G</span> is by forming the direct sum <span class="SimpleMath">L(G) = G/γ_2G ⊕ γ_2G/γ_3G ⊕ γ_3G/γ_4G ⊕ ⋯</span> of the quotients of the lower central series <span class="SimpleMath">γ_1G=G</span>, <span class="SimpleMath">γ_n+1G=[γ_nG,G]</span>. Commutation in <span class="SimpleMath">G</span> induces a Lie bracket <span class="SimpleMath">L(G)× L(G) → L(G)</span>.</p>
<p>The homology <span class="SimpleMath">H_n(L)</span> of a Lie algebra (with trivial coefficients) can be calculated as the homology of the Chevalley-Eilenberg chain complex <span class="SimpleMath">C_∗(L)</span>. This chain complex is implemented in <strong class="button">HAP</strong> in the cases where the underlying additive group of <span class="SimpleMath">L</span> is either finitely generated torsion free or finitely generated of prime exponent <span class="SimpleMath">p</span>. In these two cases the ground ring for the Lie algebra/ Chevalley-Eilenberg complex is taken to be <span class="SimpleMath">Z</span> and <span class="SimpleMath">Z_p</span> respectively.</p>
<p>For example, consider the quotient <span class="SimpleMath">G=F/γ_8F</span> of the free group <span class="SimpleMath">F=F(x,y)</span> on two generators by eighth term of its lower central series. So <span class="SimpleMath">G</span> is the <em>free nilpotent group of class 7 on two generators</em>. The following commands compute <span class="SimpleMath">H_4(L(G)) = Z_2^77 ⊕ Z_6^8 ⊕ Z_12^51 ⊕ Z_132^11 ⊕ Z^2024</span> and show that the fourth homology in this case contains 2-, 3- and 11-torsion. (The commands take an hour or so to complete.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(2),7));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=LowerCentralSeriesLieAlgebra(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=LieAlgebraHomology(L,4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected(h);</span>
[ [ 0, 2024 ], [ 2, 77 ], [ 6, 8 ], [ 12, 51 ], [ 132, 11 ] ]
</pre></div>
<p>For a free nilpotent group <span class="SimpleMath">G</span> the additive homology <span class="SimpleMath">H_n(L(G))</span> of the Lie algebra can be computed more quickly in <strong class="button">HAP</strong> than the integral group homology <span class="SimpleMath">H_n(G, Z)</span>. Clearly there are isomorphisms<span class="SimpleMath">H_1(G) ≅ H_1(L(G)) ≅ G_ab</span> of abelian groups in homological degree <span class="SimpleMath">n=1</span>. Hopf's formula can be used to establish an isomorphism <span class="SimpleMath">H_2(G) ≅ H_2(L(G))</span> also in degree <span class="SimpleMath">n=2</span>. The following two theorems provide further isomorphisms that allow for the homology of a free nilpotent group to be calculated more efficiently as the homology of the associated Lie algebra.</p>
<p><strong class="button">Theorem 1.</strong> <a href="chapBib.html#biBkuzmin">[KS98]</a> <em>Let <span class="SimpleMath">G</span> be a finitely generated free nilpotent group of class 2. Then the integral group homology <span class="SimpleMath">H_n(G, Z)</span> is isomorphic to the integral Lie algebra homology <span class="SimpleMath">H_n(L(G), Z)</span> in each degree <span class="SimpleMath">n≥0</span>.</em></p>
<p><strong class="button">Theorem 2.</strong> <a href="chapBib.html#biBigusa">[IO01]</a> <em>Let <span class="SimpleMath">G</span> be a finitely generated free nilpotent group (of any class). Then the integral group homology <span class="SimpleMath">H_n(G, Z)</span> is isomorphic to the integral Lie algebra homology <span class="SimpleMath">H_n(L(G), Z)</span> in degrees <span class="SimpleMath">n=0, 1, 2, 3</span>.</em></p>
<p>We should remark that experimentation on free nilpotent groups of class <span class="SimpleMath">≥ 4</span> has not yielded a group for which the isomorphism <span class="SimpleMath">H_n(G, Z) ≅ H_n(L(G), G)</span> fails. For instance, the isomorphism holds in degree <span class="SimpleMath">n=4</span> for the free nilpotent group of class 5 on two generators, and for the free nilpotent group of class 2 on four generators:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(2),5));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=LowerCentralSeriesLieAlgebra(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected( LieAlgebraHomology(L,4) );</span>
[ [ 0, 85 ], [ 7, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected( GroupHomology(G,4) );</span>
[ [ 0, 85 ], [ 7, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=Image(NqEpimorphismNilpotentQuotient(FreeGroup(4),2));; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=LowerCentralSeriesLieAlgebra(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected( LieAlgebraHomology(L,4) );</span>
[ [ 0, 84 ], [ 3, 4 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected( GroupHomology(G,4) );</span>
[ [ 0, 84 ], [ 3, 4 ] ]
</pre></div>
<p><a id="X7C3DEDD57BB4D537" name="X7C3DEDD57BB4D537"></a></p>
<h4>7.8 <span class="Heading">Cohomology with coefficients in a module</span></h4>
<p>There are various ways to represent a <span class="SimpleMath">ZG</span>-module <span class="SimpleMath">A</span> with action <span class="SimpleMath">G× A → A, (g,a)↦ α(g,a)</span>.</p>
<p>One possibility is to use the data type of a <em><span class="SimpleMath">G</span>-Outer Group</em> which involves three components: an <span class="SimpleMath">ActedGroup</span> <span class="SimpleMath">A</span>; an <span class="SimpleMath">Acting Group</span> <span class="SimpleMath">G</span>; a <span class="SimpleMath">Mapping</span> <span class="SimpleMath">(g,a)↦ α(g,a)</span>. The following example uses this data type to compute the cohomology <span class="SimpleMath">H^4(G,A) = Z_5 ⊕ Z_10</span> of the symmetric group <span class="SimpleMath">G=S_6</span> with coefficients in the integers <span class="SimpleMath">A= Z</span> where odd permutations act non-trivially on <span class="SimpleMath">A</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SymmetricGroup(6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=AbelianPcpGroup([0]);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">alpha:=function(g,a); return a^SignPerm(g); end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=GModuleAsGOuterGroup(G,A,alpha);</span>
ZG-module with abelian invariants [ 0 ] and G= SymmetricGroup( [ 1 .. 6 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionFiniteGroup(G,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=HomToGModule(R,A);</span>
G-cocomplex of length 5 .
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology(C,4);</span>
[ 2, 2, 5 ]
</pre></div>
<p>If <span class="SimpleMath">A= Z^n</span> and <span class="SimpleMath">G</span> acts as</p>
<p><span class="SimpleMath">G× A → A, (g, v) ↦ ρ(g) v</span></p>
<p>where <span class="SimpleMath">ρ: G→ Gl_n( Z)</span> is a (not necessarily faithful) matrix representation of degree <span class="SimpleMath">n</span> then we can avoid the use of <span class="SimpleMath">G</span>-outer groups and use just the homomorphism <span class="SimpleMath">ρ</span> instead. The following example uses this data type to compute the cohomology</p>
<p><span class="SimpleMath">H^6(G,A) = Z_2</span></p>
<p>and the homology</p>
<p><span class="SimpleMath">H_6(G,A) = 0</span></p>
<p>of the alternating group <span class="SimpleMath">G=A_5</span> with coefficients in <span class="SimpleMath">A= Z^5</span> where elements of <span class="SimpleMath">G</span> act on <span class="SimpleMath">Z^5</span> via an irreducible representation.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=AlternatingGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rho:=IrreducibleRepresentations(G)[5];</span>
[ (1,2,3,4,5), (3,4,5) ] ->
[
[ [ 0, 0, 1, 0, 0 ], [ -1, -1, 0, 0, 1 ], [ 0, 1, 1, 1, 0 ],
[ 1, 0, -1, 0, -1 ], [ -1, -1, 0, -1, 0 ] ],
[ [ -1, -1, 0, 0, 1 ], [ 1, 0, -1, 0, -1 ], [ 0, 0, 0, 0, 1 ],
[ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionFiniteGroup(G,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=HomToIntegralModule(R,rho);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology(C,6);</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=TensorWithIntegralModule(R,rho);</span>
Chain complex of length 7 in characteristic 0 .
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(D,6);</span>
[ ]
</pre></div>
<p>If <span class="SimpleMath">V=K^d</span> is a vetor space of dimension <span class="SimpleMath">d</span> over the field <span class="SimpleMath">K=GF(p)</span> with <span class="SimpleMath">p</span> a prime and <span class="SimpleMath">G</span> acts on <span class="SimpleMath">V</span> via a homomorphism <span class="SimpleMath">ρ: G→ GL_d(K)</span> then the homology <span class="SimpleMath">H^n(G,V)</span> can again be computed without the use of G-outer groups. As an example, the following commands compute</p>
<p><span class="SimpleMath">H^4(GL(3,2),V) =K^2</span></p>
<p>where <span class="SimpleMath">K=GF(2)</span> and <span class="SimpleMath">GL(3,2)</span> acts with its natural action on <span class="SimpleMath">V=K^3</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=GL(3,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rho:=GroupHomomorphismByFunction(G,G,x->x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionFiniteGroup(G,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=HomToModPModule(R,rho);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Cohomology(C,4);</span>
2
</pre></div>
<p>It can be computationally difficult to compute resolutions for large finite groups. But the <span class="SimpleMath">p</span>-primary part of the homology can be computed using resolutions of Sylow <span class="SimpleMath">p</span>-subgroups. This approach is used in the following example that computes the <span class="SimpleMath">2</span>-primary part</p>
<p><span class="SimpleMath">H_2(G, Z)_(2) = Z_2 ⊕ Z_2⊕ Z_2</span></p>
<p>of the degree 2 integral homology of the Rubik's cube group <span class="SimpleMath">G</span>. This group has order <span class="SimpleMath">43252003274489856000</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= [</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27),</span>
<span class="GAPprompt">></span> <span class="GAPinput"> (41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)</span>
<span class="GAPprompt">></span> <span class="GAPinput"> ];; G:=Group(gens);;P:=SylowSubgroup(G,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionNormalSeries(BigStepUCS(P,6),3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimePartDerivedFunctorViaSubgroupChain(G,R,TensorWithIntegers,2);</span>
[ 2, 2, 2 ]
</pre></div>
<p>The same approach is used in the following example that computes the <span class="SimpleMath">2</span>-primary part</p>
<p><span class="SimpleMath">H_11(A_7,A)_(2) = Z_2 ⊕ Z_2⊕ Z_4</span></p>
<p>of the degree 11 homology of the alternating group <span class="SimpleMath">A_7</span> of degree <span class="SimpleMath">7</span> with coefficients in the module <span class="SimpleMath">A= Z^7</span> on which <span class="SimpleMath">A_7</span> acts by permuting basis vectors.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=AlternatingGroup(7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rho:=PermToMatrixGroup(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionFiniteGroup(SylowSubgroup(G,2),12);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=function(X); return TensorWithIntegralModule(X,rho); end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimePartDerivedFunctorViaSubgroupChain(G,R,F,11);</span>
[ 2, 2, 4 ]
</pre></div>
<p>Similar commands compute</p>
<p><span class="SimpleMath">H_3(A_10,A)_(2) = Z_4</span></p>
<p>with coefficient module <span class="SimpleMath">A= Z^10</span> on which <span class="SimpleMath">A_10</span> acts by permuting basis vectors.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=AlternatingGroup(10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rho:=PermToMatrixGroup(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionFiniteGroup(SylowSubgroup(G,2),4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=function(X); return TensorWithIntegralModule(X,rho); end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimePartDerivedFunctorViaSubgroupChain(G,R,F,3);</span>
[ 4 ]
</pre></div>
<p>The following commands compute</p>
<p><span class="SimpleMath">H_100(GL(3,2),V)= K^34</span></p>
<p>where <span class="SimpleMath">V</span> is the vector space of dimension <span class="SimpleMath">3</span> over <span class="SimpleMath">K=GF(2)</span> acting via some irreducible representation <span class="SimpleMath">ρ: GL(3,2) → GL(V)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=GL(3,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">rho:=IrreducibleRepresentations(G,GF(2))[3];</span>
CompositionMapping( [ (5,7)(6,8), (2,3,5)(4,7,6) ] ->
[ <an immutable 3x3 matrix over GF2>, <an immutable 3x3 matrix over GF2> ],
<action isomorphism> )
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=function(X); return TensorWithModPModule(X,rho); end;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">S:=ResolutionPrimePowerGroup(SylowSubgroup(G,2),101);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PrimePartDerivedFunctorViaSubgroupChain(G,S,F,100);</span>
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2, 2, 2, 2, 2, 2 ]
</pre></div>
<p><a id="X7E573EA582CCEF2E" name="X7E573EA582CCEF2E"></a></p>
<h4>7.9 <span class="Heading">Cohomology as a functor of the first variable</span></h4>
<p>Suppose given a group homomorphism <span class="SimpleMath">f: G_1→ G_2</span> and a <span class="SimpleMath">G_2</span>-module <span class="SimpleMath">A</span>. Then <span class="SimpleMath">A</span> is naturally a <span class="SimpleMath">G_1</span>-module with action via <span class="SimpleMath">f</span>, and there is an induced cohomology homomorphism <span class="SimpleMath">H^n(f,A): H^n(G_2,A) → H^n(G_1,A)</span>.</p>
<p>The following example computes this cohomology homomorphism in degree <span class="SimpleMath">n=6</span> for the inclusion <span class="SimpleMath">f: A_5 → S_5</span> and <span class="SimpleMath">A= Z^5</span> with action that permutes the canonical basis. The final commands determine that the kernel of the homomorphism <span class="SimpleMath">H^6(f,A)</span> is the Klein group of order <span class="SimpleMath">4</span> and that the cokernel is cyclic of order <span class="SimpleMath">6</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G1:=AlternatingGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G2:=SymmetricGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=GroupHomomorphismByFunction(G1,G2,x->x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">pi:=PermToMatrixGroup(G2,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R1:=ResolutionFiniteGroup(G1,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R2:=ResolutionFiniteGroup(G2,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=EquivariantChainMap(R1,R2,f);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=HomToIntegralModule(F,pi);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:=Cohomology(C,6);</span>
[ g1, g2, g3 ] -> [ id, id, g3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(Kernel(c));</span>
[ 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AbelianInvariants(Range(c)/Image(c));</span>
[ 2, 3 ]
</pre></div>
<p><a id="X796731727A7EBE59" name="X796731727A7EBE59"></a></p>
<h4>7.10 <span class="Heading">Cohomology as a functor of the second variable and the long exact coefficient sequence</span></h4>
<p>A short exact sequence of <span class="SimpleMath">ZG</span>-modules <span class="SimpleMath">A ↣ B ↠ C</span> induces a long exact sequence of cohomology groups</p>
<p><span class="SimpleMath">→ H^n(G,A) → H^n(G,B) → H^n(G,C) → H^n+1(G,A) →</span> .</p>
<p>Consider the symmetric group <span class="SimpleMath">G=S_4</span> and the sequence <span class="SimpleMath">Z_4 ↣ Z_8 ↠ Z_2</span> of trivial <span class="SimpleMath">ZG</span>-modules. The following commands compute the induced cohomology homomorphism</p>
<p><span class="SimpleMath">f: H^3(S_4, Z_4) → H^3(S_4, Z_8)</span></p>
<p>and determine that the image of this induced homomorphism has order <span class="SimpleMath">8</span> and that its kernel has order <span class="SimpleMath">2</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SymmetricGroup(4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=(1,2,3,4,5,6,7,8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">a:=Group(x^2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">b:=Group(x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ahomb:=GroupHomomorphismByFunction(a,b,y->y);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=TrivialGModuleAsGOuterGroup(G,a);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:=TrivialGModuleAsGOuterGroup(G,b);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi:=GOuterGroupHomomorphism();;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi!.Source:=A;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi!.Target:=B;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">phi!.Mapping:=ahomb;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Hphi:=CohomologyHomomorphism(phi,3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(ImageOfGOuterGroupHomomorphism(Hphi));</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(KernelOfGOuterGroupHomomorphism(Hphi));</span>
2
</pre></div>
<p>The following commands then compute the homomorphism</p>
<p><span class="SimpleMath">H^3(S_4, Z_8) → H^3(S_4, Z_2)</span></p>
<p>induced by <span class="SimpleMath">Z_4 ↣ Z_8 ↠ Z_2</span>, and determine that the kernel of this homomorphsim has order <span class="SimpleMath">8</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">bhomc:=NaturalHomomorphismByNormalSubgroup(b,a);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:=TrivialGModuleAsGOuterGroup(G,b);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=TrivialGModuleAsGOuterGroup(G,Image(bhomc));</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">psi:=GOuterGroupHomomorphism();</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">psi!.Source:=B;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">psi!.Target:=C;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">psi!.Mapping:=bhomc;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Hpsi:=CohomologyHomomorphism(psi,3);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(KernelOfGOuterGroupHomomorphism(Hpsi));</span>
8
</pre></div>
<p>The following commands then compute the connecting homomorphism</p>
<p><span class="SimpleMath">H^2(S_4, Z_2) → H^3(S_4, Z_4)</span></p>
<p>and determine that the image of this homomorphism has order <span class="SimpleMath">2</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">delta:=ConnectingCohomologyHomomorphism(psi,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size(ImageOfGOuterGroupHomomorphism(delta));</span>
</pre></div>
<p>Note that the various orders are consistent with exactness of the sequence</p>
<p><span class="SimpleMath">H^2(S_4, Z_2) → H^3(S_4, Z_4) → H^3(S_4, Z_8) → H^3(S_4, Z_2)</span> .</p>
<p><a id="X80F6FD3E7C7E4E8D" name="X80F6FD3E7C7E4E8D"></a></p>
<h4>7.11 <span class="Heading">Transfer Homomorphism</span></h4>
<p>Consider the action of the symmetric group <span class="SimpleMath">G=S_5</span> on <span class="SimpleMath">A= Z^5</span> which permutes the canonical basis. The action restricts to the sylow <span class="SimpleMath">2</span>-subgroup <span class="SimpleMath">P=Syl_2(G)</span>. The following commands compute the cohomology transfer homomorphism <span class="SimpleMath">t^4: H^4(P,A) → H^4(S_5,A)</span> and determine its kernel and image. The integral homology transfer <span class="SimpleMath">t_4: H_4(S_5, Z) → H_5(P, Z)</span> is also computed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SymmetricGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">P:=SylowSubgroup(G,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionFiniteGroup(G,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=PermToMatrixGroup(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tr:=TransferCochainMap(R,P,A);</span>
Cochain Map between complexes of length 5 .
<span class="GAPprompt">gap></span> <span class="GAPinput">t4:=Cohomology(tr,4);</span>
[ g1, g2, g3, g4 ] -> [ id, g1, g2, g4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(Kernel(t4));</span>
"C2 x C2"
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(Image(t4));</span>
"C4 x C2"
<span class="GAPprompt">gap></span> <span class="GAPinput">tr:=TransferChainMap(R,P);</span>
Chain Map between complexes of length 5 .
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(tr,4);</span>
[ g1 ] -> [ g1 ]
</pre></div>
<p><a id="X79B1406C803FF178" name="X79B1406C803FF178"></a></p>
<h4>7.12 <span class="Heading">Cohomology rings of finite fundamental groups of 3-manifolds
</span></h4>
<p>A <em>spherical 3-manifold</em> is a 3-manifold arising as the quotient <span class="SimpleMath">S^3/Γ</span> of the 3-sphere <span class="SimpleMath">S^3</span> by a finite subgroup <span class="SimpleMath">Γ</span> of <span class="SimpleMath">SO(4)</span> acting freely as rotations. The geometrization conjecture, proved by Grigori Perelman, implies that every closed connected 3-manifold with a finite fundamental group is homeomorphic to a spherical 3-manifold.</p>
<p>A spherical 3-manifold <span class="SimpleMath">S^3/Γ</span> has finite fundamental group isomorphic to <span class="SimpleMath">Γ</span>. This fundamental group is one of:</p>
<ul>
<li><p><span class="SimpleMath">Γ=C_m=⟨ x | x^m⟩</span> (<strong class="button">cyclic fundamental group</strong>)</p>
</li>
<li><p><span class="SimpleMath">Γ=C_m× ⟨ x,y | xyx^-1=y^-1, x^2^k=y^n ⟩</span> for integers <span class="SimpleMath">k, m≥ 1, n≥ 2</span> and <span class="SimpleMath">m</span> coprime to <span class="SimpleMath">2n</span> (<strong class="button">prism manifold case</strong>)</p>
</li>
<li><p><span class="SimpleMath">Γ= C_m× ⟨ x,y, z | (xy)^2=x^2=y^2, zxz^-1=y, zyz^-1=xy, z^3^k=1⟩</span> for integers <span class="SimpleMath">k,m≥ 1</span> and <span class="SimpleMath">m</span> coprime to 6 (<strong class="button">tetrahedral case</strong>)</p>
</li>
<li><p><span class="SimpleMath">Γ=C_m×⟨ x,y | (xy)^2=x^3=y^4⟩</span> for <span class="SimpleMath">m≥ 1</span> coprime to 6 (<strong class="button">octahedral case</strong>)</p>
</li>
<li><p><span class="SimpleMath">Γ=C_m× ⟨ x,y | (xy)^2=x^3=y^5⟩</span> for <span class="SimpleMath">m≥ 1</span> coprime to 30 (<strong class="button">icosahedral case</strong>).</p>
</li>
</ul>
<p>This list of cases is taken from the <span class="URL"><a href="https://en.wikipedia.org/wiki/Spherical_3-manifold">Wikipedia pages</a></span>. The group <span class="SimpleMath">Γ</span> has periodic cohomology since it acts on a sphere. The cyclic group has period 2 and in the other four cases it has period 4. (Recall that in general a finite group <span class="SimpleMath">G</span> has <em>periodic cohomology of period <span class="SimpleMath">n</span></em> if there is an element <span class="SimpleMath">u∈ H^n(G, Z)</span> such that the cup product <span class="SimpleMath">- ∪ u: H^k(G, Z) → H^k+n(G, Z)</span> is an isomorphism for all <span class="SimpleMath">k≥ 1</span>. It can be shown that <span class="SimpleMath">G</span> has periodic cohomology of period <span class="SimpleMath">n</span> if and only if <span class="SimpleMath">H^n(G, Z)= Z_|G|</span>.)</p>
<p>The cohomology of the cyclic group is well-known, and the cohomology of a direct product can be obtained from that of the factors using the Kunneth formula.</p>
<p>In the icosahedral case with <span class="SimpleMath">m=1</span> the following commands yield $$H^\ast(\Gamma,\mathbb Z)=Z[t]/(120t=0)$$ with generator <span class="SimpleMath">t</span> of degree 4. The final command demonstrates that a periodic resolution is used in the computation.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup(2);;x:=F.1;;y:=F.2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=F/[(x*y)^2*x^-3, x^3*y^-5];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(G);</span>
120
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionSmallGroup(G,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=0;;Cohomology(HomToIntegers(R),n);</span>
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=1;;Cohomology(HomToIntegers(R),n);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=2;;Cohomology(HomToIntegers(R),n);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=3;;Cohomology(HomToIntegers(R),n);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=4;;Cohomology(HomToIntegers(R),n);</span>
[ 120 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..5],k->R!.dimension(k));</span>
[ 1, 2, 2, 1, 1, 2 ]
</pre></div>
<p>In the octahedral case with <span class="SimpleMath">m=1</span> we obtain $$H^\ast(\Gamma,\mathbb Z) = \mathbb Z[s,t]/(s^2=24t, 2s=0, 48t=0)$$ where <span class="SimpleMath">s</span> has degree 2 and <span class="SimpleMath">t</span> has degree 4, from the following commands.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup(2);;x:=F.1;;y:=F.2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=F/[(x*y)^2*x^-3, x^3*y^-4];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(G);</span>
48
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionFiniteGroup(G,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=0;;Cohomology(HomToIntegers(R),n);</span>
[ 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=1;;Cohomology(HomToIntegers(R),n);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=2;;Cohomology(HomToIntegers(R),n);</span>
[ 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=3;;Cohomology(HomToIntegers(R),n);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=4;;Cohomology(HomToIntegers(R),n);</span>
[ 48 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IntegralCupProduct(R,[1],[1],2,2);</span>
[ 24 ]
</pre></div>
<p>In the tetrahedral case with <span class="SimpleMath">m=1</span> we obtain $$H^\ast(\Gamma,\mathbb Z) = \mathbb Z[s,t]/(s^2=16t, 3s=0, 24t=0)$$ where <span class="SimpleMath">s</span> has degree 2 and <span class="SimpleMath">t</span> has degree 4, from the following commands.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=FreeGroup(3);;x:=F.1;;y:=F.2;;z:=F.3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=F/[(x*y)^2*x^-2, x^2*y^-2, z*x*z^-1*y^-1, z*y*z^-1*y^-1*x^-1,z^3];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Order(G);</span>
24
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionFiniteGroup(G,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=1;;Cohomology(HomToIntegers(R),n);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=2;;Cohomology(HomToIntegers(R),n);</span>
[ 3 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=3;;Cohomology(HomToIntegers(R),n);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">n:=4;;Cohomology(HomToIntegers(R),n);</span>
[ 24 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IntegralCupProduct(R,[1],[1],2,2);</span>
[ 16 ]
</pre></div>
<p>A theoretical calculation of the integral and mod-p cohomology rings of all of these fundamental groups of spherical 3-manifolds is given in <a href="chapBib.html#biBtomoda">[TZ08]</a>.</p>
<p><a id="X833A19F0791C3B06" name="X833A19F0791C3B06"></a></p>
<h4>7.13 <span class="Heading">Explicit cocycles </span></h4>
<p>Given a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R_∗</span> and a <span class="SimpleMath">ZG</span>-module <span class="SimpleMath">A</span>, one defines an <em><span class="SimpleMath">n</span>-cocycle</em> to be a <span class="SimpleMath">ZG</span>-homomorphism <span class="SimpleMath">f: R_n → A</span> for which the composite homomorphism <span class="SimpleMath">fd_n+1: R_n+1→ A</span> is zero. If <span class="SimpleMath">R_∗</span> happens to be the standard bar resolution (i.e. the cellular chain complex of the nerve of the group <span class="SimpleMath">G</span> considered as a one object category) then the free <span class="SimpleMath">ZG</span>-generators of <span class="SimpleMath">R_n</span> are indexed by <span class="SimpleMath">n</span>-tuples <span class="SimpleMath">(g_1 | g_2 | ... | g_n)</span> of elements <span class="SimpleMath">g_i</span> in <span class="SimpleMath">G</span>. In this case we say that the <span class="SimpleMath">n</span>-cocycle is a <em>standard n-cocycle</em> and we think of it as a set-theoretic function</p>
<p><span class="SimpleMath">f : G × G × ⋯ × G ⟶ A</span></p>
<p>satisfying a certain algebraic cocycle condition. Bearing in mind that a standard <span class="SimpleMath">n</span>-cocycle really just assigns an element <span class="SimpleMath">f(g_1, ... ,g_n) ∈ A</span> to an <span class="SimpleMath">n</span>-simplex in the nerve of <span class="SimpleMath">G</span> , the cocycle condition is a very natural one which states that <em><span class="SimpleMath">f</span> must vanish on the boundary of a certain <span class="SimpleMath">(n+1)</span>-simplex</em>. For <span class="SimpleMath">n=2</span> the condition is that a <span class="SimpleMath">2</span>-cocycle <span class="SimpleMath">f(g_1,g_2)</span> must satisfy</p>
<p><span class="SimpleMath">g.f(h,k) + f(g,hk) = f(gh,k) + f(g,h)</span></p>
<p>for all <span class="SimpleMath">g,h,k ∈ G</span>. This equation is explained by the following picture.</p>
<p><img src="images/cocycle.png" align="center" height="200" alt="2-cocycle equation"/></p>
<p>The definition of a cocycle clearly depends on the choice of <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R_∗</span>. However, the cohomology group <span class="SimpleMath">H^n(G,A)</span>, which is a group of equivalence classes of <span class="SimpleMath">n</span>-cocycles, is independent of the choice of <span class="SimpleMath">R_∗</span>.</p>
<p>There are some occasions when one needs explicit examples of standard cocycles. For instance:</p>
<ul>
<li><p>Let <span class="SimpleMath">G</span> be a finite group and <span class="SimpleMath">k</span> a field of characteristic <span class="SimpleMath">0</span>. The group algebra <span class="SimpleMath">k(G)</span>, and the algebra <span class="SimpleMath">F(G)</span> of functions <span class="SimpleMath">d_g: G→ k, h→ d_g,h</span>, are both Hopf algebras. The tensor product <span class="SimpleMath">F(G) ⊗ k(G)</span> also admits a Hopf algebra structure known as the quantum double <span class="SimpleMath">D(G)</span>. A twisted quantum double <span class="SimpleMath">D_f(G)</span> was introduced by R. Dijkraaf, V. Pasquier & P. Roche <a href="chapBib.html#biBdpr">[DPR91]</a>. The twisted double is a quasi-Hopf algebra depending on a <span class="SimpleMath">3</span>-cocycle <span class="SimpleMath">f: G× G× G→ k</span>. The multiplication is given by <span class="SimpleMath">(d_g ⊗ x)(d_h ⊗ y) = d_gx,xhβ_g(x,y)(d_g ⊗ xy)</span> where <span class="SimpleMath">β_a</span> is defined by <span class="SimpleMath">β_a(h,g) = f(a,h,g) f(h,h^-1ah,g)^-1 f(h,g,(hg)^-1ahg)</span> . Although the algebraic structure of <span class="SimpleMath">D_f(G)</span> depends very much on the particular <span class="SimpleMath">3</span>-cocycle <span class="SimpleMath">f</span>, representation-theoretic properties of <span class="SimpleMath">D_f(G)</span> depend only on the cohomology class of <span class="SimpleMath">f</span>.</p>
</li>
<li><p>An explicit <span class="SimpleMath">2</span>-cocycle <span class="SimpleMath">f: G× G→ A</span> is needed to construct the multiplication <span class="SimpleMath">(a,g)(a',g') = (a + g⋅ a' + f(g,g'), gg')</span> in the extension a group <span class="SimpleMath">G</span> by a <span class="SimpleMath">ZG</span>-module <span class="SimpleMath">A</span> determined by the cohomology class of <span class="SimpleMath">f</span> in <span class="SimpleMath">H^2(G,A)</span>. See <a href="chap6.html#X8333413B838D787D"><span class="RefLink">6.7</span></a>.</p>
</li>
<li><p>In work on coding theory and Hadamard matrices a number of papers have investigated square matrices <span class="SimpleMath">(a_ij)</span> whose entries <span class="SimpleMath">a_ij=f(g_i,g_j)</span> are the values of a <span class="SimpleMath">2</span>-cocycle <span class="SimpleMath">f: G× G → Z_2</span> where <span class="SimpleMath">G</span> is a finite group acting trivially on <span class="SimpleMath">Z_2</span>. See for instance <a href="chapBib.html#biBhoradam">[Hor00]</a> and <a href="chap6.html#X7C60E2B578074532"><span class="RefLink">6.10</span></a>.</p>
</li>
</ul>
<p>Given a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R_∗</span> (with contracting homotopy) and a <span class="SimpleMath">ZG</span>-module <span class="SimpleMath">A</span> one can use HAP commands to compute explicit standard <span class="SimpleMath">n</span>-cocycles <span class="SimpleMath">f: G^n → A</span>. With the twisted quantum double in mind, we illustrate the computation for <span class="SimpleMath">n=3</span>, <span class="SimpleMath">G=S_3</span>, and <span class="SimpleMath">A=U(1)</span> the group of complex numbers of modulus <span class="SimpleMath">1</span> with trivial <span class="SimpleMath">G</span>-action.</p>
<p>We first compute a <span class="SimpleMath">ZG</span>-resolution <span class="SimpleMath">R_∗</span>. The Universal Coefficient Theorem gives an isomorphism <span class="SimpleMath">H_3(G,U(1)) = Hom_ Z(H_3(G, Z), U(1))</span>, The multiplicative group <span class="SimpleMath">U(1)</span> can thus be viewed as <span class="SimpleMath">Z_m</span> where <span class="SimpleMath">m</span> is a multiple of the exponent of <span class="SimpleMath">H_3(G, Z)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SymmetricGroup(3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionFiniteGroup(G,4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TR:=TensorWithIntegers(R);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(TR,3);</span>
[ 6 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">R!.dimension(3);</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">R!.dimension(4);</span>
5
</pre></div>
<p>We thus replace the very infinite group U(1) by the finite cyclic group <span class="SimpleMath">Z_6</span>. Since the resolution <span class="SimpleMath">R_∗</span> has <span class="SimpleMath">4</span> generators in degree <span class="SimpleMath">3</span>, a homomorphism <span class="SimpleMath">f: R^3→ U(1)</span> can be represented by a list <span class="SimpleMath">f=[f_1, f_2, f_3, f_4]</span> with <span class="SimpleMath">f_i</span> the image in <span class="SimpleMath">Z_6</span> of the <span class="SimpleMath">i</span>th generator. The cocycle condition on <span class="SimpleMath">f</span> can be expressed as a matrix equation</p>
<p><span class="SimpleMath">Mf^t = 0 mod 6</span>.</p>
<p>where the matrix <span class="SimpleMath">M</span> is obtained from the following command and <span class="SimpleMath">f^t</span> denotes the transpose.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=CocycleCondition(R,3);;</span>
</pre></div>
<p>A particular cocycle <span class="SimpleMath">f=[f_1, f_2, f_3, f_4]</span> can be obtained by choosing a solution to the equation Mf^t=0.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SolutionsMod2:=NullspaceModQ(TransposedMat(M),2);</span>
[ [ 0, 0, 0, 0 ], [ 0, 0, 1, 1 ], [ 1, 1, 0, 0 ], [ 1, 1, 1, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">SolutionsMod3:=NullspaceModQ(TransposedMat(M),3);</span>
[ [ 0, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 2 ], [ 0, 0, 1, 0 ],
[ 0, 0, 1, 1 ], [ 0, 0, 1, 2 ], [ 0, 0, 2, 0 ], [ 0, 0, 2, 1 ],
[ 0, 0, 2, 2 ] ]
</pre></div>
<p>A non-standard <span class="SimpleMath">3</span>-cocycle <span class="SimpleMath">f</span> can be converted to a standard one using the command <code class="code">StandardCocycle(R,f,n,q)</code> . This command inputs <span class="SimpleMath">R_∗</span>, integers <span class="SimpleMath">n</span> and <span class="SimpleMath">q</span>, and an <span class="SimpleMath">n</span>-cocycle <span class="SimpleMath">f</span> for the resolution <span class="SimpleMath">R_∗</span>. It returns a standard cocycle <span class="SimpleMath">G^n → Z_q</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=3*SolutionsMod2[3] - SolutionsMod3[5]; #An example solution to Mf=0 mod 6.</span>
[ 3, 3, -1, -1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Standard_f:=StandardCocycle(R,f,3,6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Random(G); h:=Random(G); k:=Random(G);</span>
(1,2)
(1,3,2)
(1,3)
<span class="GAPprompt">gap></span> <span class="GAPinput">Standard_f(g,h,k);</span>
3
</pre></div>
<p>A function <span class="SimpleMath">f: G× G× G → A</span> is a standard <span class="SimpleMath">3</span>-cocycle if and only if</p>
<p><span class="SimpleMath">g⋅ f(h,k,l) - f(gh,k,l) + f(g,hk,l) - f(g,h,kl) + f(g,h,k) = 0</span></p>
<p>for all <span class="SimpleMath">g,h,k,l ∈ G</span>. In the above example the group <span class="SimpleMath">G=S_3</span> acts trivially on <span class="SimpleMath">A=Z_6</span>. The following commands show that the standard <span class="SimpleMath">3</span>-cocycle produced in the example really does satisfy this <span class="SimpleMath">3</span>-cocycle condition.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">sf:=Standard_f;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Test:=function(g,h,k,l);</span>
<span class="GAPprompt">></span> <span class="GAPinput">return sf(h,k,l) - sf(g*h,k,l) + sf(g,h*k,l) - sf(g,h,k*l) + sf(g,h,k);</span>
<span class="GAPprompt">></span> <span class="GAPinput">end;</span>
function( g, h, k, l ) ... end
<span class="GAPprompt">gap></span> <span class="GAPinput">for g in G do for h in G do for k in G do for l in G do</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print(Test(g,h,k,l),",");</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;od;od;od;</span>
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,6,6,0,0,6,
0,0,0,0,0,6,6,6,0,6,0,12,12,6,12,6,0,12,6,0,6,6,0,0,0,0,0,0,0,12,12,6,6,6,0,
6,6,0,6,6,0,0,-6,0,0,0,0,0,0,0,0,0,0,6,6,6,6,6,0,0,0,0,0,0,0,6,0,0,6,6,0,6,6,
0,6,0,0,6,6,6,0,0,0,0,0,0,0,-6,0,0,-6,0,-6,0,0,0,0,0,0,0,0,6,6,0,6,0,0,6,0,0,
0,0,0,6,6,6,0,0,0,6,6,6,0,0,0,0,-6,0,6,6,0,0,0,0,0,0,0,12,6,6,0,6,0,0,0,0,12,
6,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,0,6,0,0,0,0,0,6,6,
6,0,0,0,6,12,6,6,0,0,0,-6,0,0,6,0,0,0,0,0,0,0,12,12,6,6,6,0,0,0,0,6,6,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,0,6,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,6,6,6,6,0,
6,6,0,6,6,0,12,12,6,12,12,0,0,0,0,0,0,0,6,6,0,0,0,0,6,6,6,12,12,0,-6,-6,0,0,
0,0,6,6,0,0,6,0,0,6,0,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,6,
0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0,6,6,0,6,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,6,6,0,6,6,0,6,0,0,6,6,6,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,0,0,0,0,0,0,0,-6,0,6,0,6,0,6,0,0,0,0,0,0,0,12,12,6,12,12,0,6,6,0,6,6,0,
0,0,0,0,0,0,12,12,6,12,12,0,6,6,0,6,6,0,0,0,0,0,0,0,0,0,0,0,0,0,6,6,6,6,6,0,
0,0,0,0,0,0,6,0,0,6,6,0,6,6,0,6,0,0,6,6,6,0,0,0,-6,0,0,0,-6,0,0,-6,0,-6,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,6,6,6,6,0,6,6,0,0,0,0,0,0,0,6,6,0,0,0,
0,0,0,0,6,6,0,-6,0,0,-6,0,0,12,6,0,-6,-6,0,0,0,0,6,6,0,0,6,0,0,6,0,6,6,0,0,0,
0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
0,0,0,-6,0,0,0,0,0,0,0,0,0,0,6,6,6,6,6,0,6,12,0,6,0,0,6,0,0,0,6,0,0,0,0,0,0,
0,6,12,0,0,0,0,0,0,0,6,6,0,-6,-6,0,0,0,0,0,0,0,0,6,0,0,6,0,6,6,0,0,0,0,0,0,0,
6,0,0,0,6,0,0,6,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,6,
0,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,6,6,0,6,6,0,6,6,6,12,12,0,0,0,0,0,0,0,6,6,0,
6,6,0,6,6,6,12,12,0,0,0,0,0,0,0,6,6,0,0,6,0,0,6,0,6,6,
</pre></div>
<p><a id="X7C5233E27D2D603E" name="X7C5233E27D2D603E"></a></p>
<h4>7.14 <span class="Heading">Quillen's complex and the <span class="SimpleMath">p</span>-part of homology </span></h4>
<p>Let <span class="SimpleMath">G</span> be a finite group with order divisible by prime <span class="SimpleMath">p</span>. Let <span class="SimpleMath">mathcal A=mathcal A_p(G)</span> denote Quillen's simplicial complex arising as the order complex of the poset of non-trivial elementary abelian <span class="SimpleMath">p</span>-subgroups of <span class="SimpleMath">G</span>. The group <span class="SimpleMath">G</span> acts on <span class="SimpleMath">mathcal A</span>. Denote the orbit of a <span class="SimpleMath">k</span>-simplex <span class="SimpleMath">e^k</span> by <span class="SimpleMath">[e^k]</span>, and the stabilizer of <span class="SimpleMath">e^k</span> by <span class="SimpleMath">Stab(e^k) ≤ G</span>. For a finite abelian group <span class="SimpleMath">H</span> let <span class="SimpleMath">H_p</span> denote the Sylow <span class="SimpleMath">p</span>-subgroup or the "<span class="SimpleMath">p</span>-part". In Theorem 3.3 of <a href="chapBib.html#biBWebb">[Web87]</a> P.J. Webb proved the following.</p>
<p><strong class="button">Theorem.</strong><a href="chapBib.html#biBWebb">[Web87]</a> For any <span class="SimpleMath">G</span>-module <span class="SimpleMath">M</span> there is a (non natural) isomomorphism</p>
<p><span class="SimpleMath">H_n(G,M)_p ⊕ ⨁_[e^k] : k~ odd~H_n(Stab(e^k),M)_p ≅ ⨁_[e^k] : k~ even~H_n(Stab(e^k),M)_p</span></p>
<p>for <span class="SimpleMath">n≥ 1</span>. The isomorphism can also be expressed as</p>
<p><span class="SimpleMath">H_n(G,M)_p ≅ ⨁_[e^k] : k~ even~H_n(Stab(e^k),M)_p - ⨁_[e^k] : k~ odd~H_n(Stab(e^k),M)_p</span></p>
<p>where terms can often be cancelled.</p>
<p>Thus the additive structure of the <span class="SimpleMath">p</span>-part of the homology of <span class="SimpleMath">G</span> is determined by that of the stabilizer groups. The result also holds with homology replaced by cohomology.</p>
<p><strong class="button">Illustration 1</strong></p>
<p>As an illustration of the theorem, the following commands calculate</p>
<p><span class="SimpleMath">H_n(SL_3( Z_2), Z) ≅ H_n(S_4, Z)_2 ⊕ H_n(S_4, Z)_2 ⊖ H_n(D_8, Z)_2 ⊕ H_n(S_3, Z)_3 ⊕ H_n(C_7 : C_3, Z)_7</span></p>
<p>where <span class="SimpleMath">n≥ 1</span>, <span class="SimpleMath">S_k</span> denotes the symmetric group on <span class="SimpleMath">n</span> letters, <span class="SimpleMath">D_8</span> the dihedral group of order <span class="SimpleMath">8</span> and <span class="SimpleMath">C_7 : C_3</span> a nonabelian semi-direct product of cyclic groups. Furthermore, for <span class="SimpleMath">n≥ 1</span></p>
<p><span class="SimpleMath">H_n(C_7 : C_3, Z)_7 ={beginarrayll Z_7, n ≡ 5 mod 6 0, otherwise endarray.</span></p>
<p>and</p>
<p><span class="SimpleMath">H_n(S_3, Z)_3 ={beginarrayll Z_3, n ≡ 3 mod 4 0, n ~otherwise . endarray.</span></p>
<p>Formulas for <span class="SimpleMath">H_n(S_4, Z)</span> and <span class="SimpleMath">H_n(D_8, Z)</span> can be found in the literature. Alternatively, they can be computed using <strong class="button">GAP</strong> for a given value of <span class="SimpleMath">n</span>. For <span class="SimpleMath">n=27</span> we find</p>
<p><span class="SimpleMath">H_27(S_4, Z)_2 ⊕ H_27(S_4, Z)_2 ⊖ H_27(D_8, Z)_2 ≅ Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_4</span></p>
<p>and</p>
<p><span class="SimpleMath">H_27(SL_3( Z_2), Z) ≅ Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_2 ⊕ Z_4 ⊕ Z_3</span> .</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SL(3,2);;Factors(Order(G));</span>
[ 2, 2, 2, 3, 7 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">D2:=HomologicalGroupDecomposition(G,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D3:=HomologicalGroupDecomposition(G,3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D7:=HomologicalGroupDecomposition(G,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D2[1],StructureDescription);</span>
[ "S4", "S4" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D2[2],StructureDescription);</span>
[ "D8" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D3[1],StructureDescription);</span>
[ "S3" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D3[2],StructureDescription);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D7[1],StructureDescription);</span>
[ "C7 : C3" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D7[2],StructureDescription);</span>
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CohomologicalPeriod(D7[1][1]);</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">List([1..6],n->GroupHomology(D7[1][1],n));</span>
[ [ 3 ], [ ], [ 3 ], [ ], [ 3, 7 ], [ ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CohomologicalPeriod(D3[1][1]);</span>
4
<span class="GAPprompt">gap></span> <span class="GAPinput">List([1..4],n->GroupHomology(D3[1][1],n));</span>
[ [ 2 ], [ ], [ 6 ], [ ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">R_S4:=ResolutionFiniteGroup(Group([(1,2),(2,3),(3,4)]),28);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">R_D8:=ResolutionFiniteGroup(Group([(1,2),(1,3)(2,4)]),28);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(TensorWithIntegers(R_S4),27);</span>
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 12 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(TensorWithIntegers(R_D8),27);</span>
[ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4 ]
</pre></div>
<p><strong class="button">Illustration 2</strong></p>
<p>As a further illustration of the theorem, the following commands calculate</p>
<p><span class="SimpleMath">H_n(M_12,M)_3 ≅ ⨁_1≤ i≤ 3H_n(Stab_i,M)_3 - ⨁_4≤ i≤ 5H_n(Stab_i,M)_3</span></p>
<p>for the Mathieu simple group <span class="SimpleMath">M_12</span> of order <span class="SimpleMath">95040</span>, where</p>
<p><span class="SimpleMath">Stab_1≅ Stab_3=(((C_3 × C_3) : Q_8) : C_3) : C_2</span></p>
<p><span class="SimpleMath">Stab_2=A_4 × S_3</span></p>
<p><span class="SimpleMath">Stab_4=C_3 × S_3</span></p>
<p><span class="SimpleMath">Stab_5=((C_3 × C_3) : C_3) : (C_2 × C_2)</span> .</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=MathieuGroup(12);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=HomologicalGroupDecomposition(G,3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D[1],StructureDescription);</span>
[ "(((C3 x C3) : Q8) : C3) : C2", "A4 x S3", "(((C3 x C3) : Q8) : C3) : C2" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D[2],StructureDescription);</span>
[ "C3 x S3", "((C3 x C3) : C3) : (C2 x C2)" ]
</pre></div>
<p><strong class="button">Illustration 3</strong></p>
<p>As a third illustration, the following commands show that <span class="SimpleMath">H_n(M_23,M)_p</span> is periodic for primes <span class="SimpleMath">p=5, 7, 11, 23</span> of periods dividing <span class="SimpleMath">8, 6, 10, 22</span> respectively.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=MathieuGroup(23);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Factors(Order(G));</span>
[ 2, 2, 2, 2, 2, 2, 2, 3, 3, 5, 7, 11, 23 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sd:=StructureDescription;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=HomologicalGroupDecomposition(G,5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D[1],sd);List(D[2],sd);</span>
[ "C15 : C4" ]
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPeriodic(D[1][1]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CohomologicalPeriod(D[1][1]);</span>
8
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=HomologicalGroupDecomposition(G,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D[1],sd);List(D[2],sd);</span>
[ "C2 x (C7 : C3)" ]
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPeriodic(D[1][1]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CohomologicalPeriod(D[1][1]);</span>
6
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=HomologicalGroupDecomposition(G,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D[1],sd);List(D[2],sd);</span>
[ "C11 : C5" ]
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPeriodic(D[1][1]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CohomologicalPeriod(D[1][1]);</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=HomologicalGroupDecomposition(G,23);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D[1],sd);List(D[2],sd);</span>
[ "C23 : C11" ]
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPeriodic(D[1][1]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">CohomologicalPeriod(D[1][1]);</span>
22
</pre></div>
<p>The order <span class="SimpleMath">|M_23|=10200960</span> is divisible by primes <span class="SimpleMath">p=2, 3, 5, 7, 11, 23</span>. For <span class="SimpleMath">p=3</span> the following commands establish that the Poincare series</p>
<p><span class="SimpleMath">(x^16 - 2x^15</span> <span class="SimpleMath">+ 3x^14 - 4x^13</span> <span class="SimpleMath">+ 4x^12 - 4x^11</span> <span class="SimpleMath">+ 4x^10 - 3x^9</span> <span class="SimpleMath">+ 3x^8 - 3x^7 +</span> <span class="SimpleMath">4x^6 - 4x^5</span> <span class="SimpleMath">+ 4x^4 -4x^3</span> <span class="SimpleMath">+ 3x^2 -2x + 1) /</span> <span class="SimpleMath">(x^18 - 2x^17</span> <span class="SimpleMath">+ 3x^16 - 4x^15</span> <span class="SimpleMath">+ 4x^14 -</span> <span class="SimpleMath">4x^13 + 4x^12</span> <span class="SimpleMath">- 4x^11 + 4x^10</span> <span class="SimpleMath">- 4x^9 + 4x^8</span> <span class="SimpleMath">- 4x^7 + 4x^6</span> <span class="SimpleMath">- 4x^5 + 4x^4</span> <span class="SimpleMath">- 4x^3 +</span> <span class="SimpleMath">3x^2 - 2x + 1)</span></p>
<p>describes the dimension of the vector space <span class="SimpleMath">H^n(M_23, Z_3)</span> up to at least degree <span class="SimpleMath">n=40</span>. To prove that it describes the dimension in all degrees one would need to verify "completion criteria".</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=MathieuGroup(23);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=HomologicalGroupDecomposition(G,3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D[1],StructureDescription);</span>
[ "(C3 x C3) : QD16", "A5 : S3" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(D[2],StructureDescription);</span>
[ "S3 x S3" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">P1:=PoincareSeriesPrimePart(D[1][1],3,40);</span>
(x_1^16-2*x_1^15+3*x_1^14-4*x_1^13+4*x_1^12-4*x_1^11+4*x_1^10-3*x_1^9+3*x_1^8-3*x_1^7+4*x_1^6-4*x_1^5+\
4*x_1^4-4*x_1^3+3*x_1^2-2*x_1+1)/(x_1^18-2*x_1^17+3*x_1^16-4*x_1^15+4*x_1^14-4*x_1^13+4*x_1^12-4*x_1^1\
1+4*x_1^10-4*x_1^9+4*x_1^8-4*x_1^7+4*x_1^6-4*x_1^5+4*x_1^4-4*x_1^3+3*x_1^2-2*x_1+1)
<span class="GAPprompt">gap></span> <span class="GAPinput">P2:=PoincareSeriesPrimePart(D[1][2],3,40);</span>
(x_1^4-2*x_1^3+3*x_1^2-2*x_1+1)/(x_1^6-2*x_1^5+3*x_1^4-4*x_1^3+3*x_1^2-2*x_1+1)
<span class="GAPprompt">gap></span> <span class="GAPinput">P3:=PoincareSeriesPrimePart(D[2][1],3,40);</span>
(x_1^4-2*x_1^3+3*x_1^2-2*x_1+1)/(x_1^6-2*x_1^5+3*x_1^4-4*x_1^3+3*x_1^2-2*x_1+1)
</pre></div>
<p><a id="X865CC8E0794C0E61" name="X865CC8E0794C0E61"></a></p>
<h4>7.15 <span class="Heading">Homology of a Lie algebra</span></h4>
<p>Let <span class="SimpleMath">A</span> be the Lie algebra constructed from the associative algebra <span class="SimpleMath">M^4× 4( Q)</span> of all <span class="SimpleMath">4× 4</span> rational matrices. Let <span class="SimpleMath">V</span> be its adjoint module (with underlying vector space of dimension <span class="SimpleMath">16</span> and equal to that of <span class="SimpleMath">A</span>). The following commands compute <span class="SimpleMath">H_4(A,V) = Q</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=FullMatrixAlgebra(Rationals,4);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=LieAlgebra(M);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">V:=AdjointModule(A);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=ChevalleyEilenbergComplex(V,17);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..17],C!.dimension);</span>
[ 16, 256, 1920, 8960, 29120, 69888, 128128, 183040, 205920, 183040, 128128,
69888, 29120, 8960, 1920, 256, 16, 0 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Homology(C,4);</span>
1
</pre></div>
<p>Note that the eighth term <span class="SimpleMath">C_8(V)</span> in the Chevalley-Eilenberg complex <span class="SimpleMath">C_∗(V)</span> is a vector space of dimension <span class="SimpleMath">205920</span> and so it will take longer to compute the homology in degree <span class="SimpleMath">8</span>.</p>
<p>As a second example, let <span class="SimpleMath">B</span> be the classical Lie ring of type <span class="SimpleMath">B_3</span> over the ring of integers. The following commands compute <span class="SimpleMath">H_3(B, Z)= Z ⊕ Z_2^105</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=SimpleLieAlgebra("B",7,Integers); </span>
<Lie algebra of dimension 105 over Integers>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=ChevalleyEilenbergComplex(A,4,"sparse");</span>
Sparse chain complex of length 4 in characteristic 0 .
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=ContractedComplex(C);</span>
Sparse chain complex of length 4 in characteristic 0 .
<span class="GAPprompt">gap></span> <span class="GAPinput">Collected(Homology(D,3));</span>
[ [ 0, 1 ], [ 2, 105 ] ]
</pre></div>
<p><a id="X86B4EE4783A244F7" name="X86B4EE4783A244F7"></a></p>
<h4>7.16 <span class="Heading">Covers of Lie algebras</span></h4>
<p>A short exact sequence of Lie algebras</p>
<p><span class="SimpleMath">M ↣ C ↠ L</span></p>
<p>(over a field <span class="SimpleMath">k</span>) is said to be a <em>stem extension</em> of <span class="SimpleMath">L</span> if <span class="SimpleMath">M</span> lies both in the centre <span class="SimpleMath">Z(C)</span> and in the derived subalgeba <span class="SimpleMath">C^2</span>. If, in addition, the rank of the vector space <span class="SimpleMath">M</span> is equal to the rank of the second Chevalley-Eilenberg homology <span class="SimpleMath">H_2(L,k)</span> then the Lie algebra <span class="SimpleMath">C</span> is said to be a <em>cover</em> of <span class="SimpleMath">L</span>.</p>
<p>Each finite dimensional Lie algebra <span class="SimpleMath">L</span> admits a cover <span class="SimpleMath">C</span>, and this cover can be shown to be unique up to Lie isomorphism.</p>
<p>The cover can be used to determine whether there exists a Lie algebra <span class="SimpleMath">E</span> whose central quotient <span class="SimpleMath">E/Z(E)</span> is isomorphic to <span class="SimpleMath">L</span>. The image in <span class="SimpleMath">L</span> of the centre of <span class="SimpleMath">C</span> is called the <em>Lie Epicentre</em> of <span class="SimpleMath">L</span>, and this image is trivial if and only if such an <span class="SimpleMath">E</span> exists.</p>
<p>The cover can also be used to determine the stem extensions of <span class="SimpleMath">L</span>. It can be shown that each stem extension is a quotient of the cover by an ideal in the Lie multiplier <span class="SimpleMath">H_2(L,k)</span>.</p>
<p><a id="X7DFF32A67FF39C82" name="X7DFF32A67FF39C82"></a></p>
<h5>7.16-1 <span class="Heading">Computing a cover</span></h5>
<p>The following commands compute the cover <span class="SimpleMath">C</span> of the solvable but non-nilpotent 13-dimensional Lie algebra <span class="SimpleMath">L</span> (over <span class="SimpleMath">k= Q</span>) that was introduced by M. Wuestner <a href="chapBib.html#biBWustner">[Wue92]</a>. They also show that: the second homology of <span class="SimpleMath">C</span> is trivial and compute the ranks of the homology groups in other dimensions; the Lie algebra <span class="SimpleMath">L</span> is not isomorphic to any central quotient <span class="SimpleMath">E/Z(E)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SCTL:=EmptySCTable(13,0,"antisymmetric");;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 1, 6, [ 1, 7 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 1, 8, [ 1, 9 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 1, 10, [ 1, 11 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 1, 12, [ 1, 13 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 1, 7, [ -1, 6 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 1, 9, [ -1, 8 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 1, 11, [ -1, 10 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 1, 13, [ -1, 12 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 6, 7, [ 1, 2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 8, 9, [ 1, 3 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 6, 9, [ -1, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 7, 8, [ 1, 5 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 2, 8, [ 1, 12 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 2, 9, [ 1, 13 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 3, 6, [ 1, 10 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 3, 7, [ 1, 11 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 2, 3, [ 1, 4 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 5, 6, [ -1, 12 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 5, 7, [ -1, 13 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 5, 8, [ -1, 10 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 5, 9, [ -1, 11 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 6, 11, [ -1/2, 4 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 7, 10, [ 1/2, 4 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 8, 13, [ 1/2, 4 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SetEntrySCTable( SCTL, 9, 12, [ -1/2, 4 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=LieAlgebraByStructureConstants(Rationals,SCTL);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=Source(LieCoveringHomomorphism(L));</span>
<Lie algebra of dimension 15 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension(LieEpiCentre(L));</span>
1
<span class="GAPprompt">gap></span> <span class="GAPinput">ch:=ChevalleyEilenbergComplex(C,17);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List([0..16],n->Homology(ch,n)); </span>
[ 1, 1, 0, 9, 23, 27, 47, 88, 88, 47, 27, 23, 9, 0, 1, 1, 0 ]
</pre></div>
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