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<div class="ChapSects"><a href="chap8_mj.html#X7ED29A58858AAAF2">8 <span class="Heading">Cohomology rings and Steenrod operations for groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X877CAF8B7E64DE04">8.1 <span class="Heading">Mod-<span class="SimpleMath">\(p\)</span> cohomology rings of finite groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X870E0299782638AF">8.1-1 <span class="Heading">Ring presentations (for the commutative <span class="SimpleMath">\(p=2\)</span> case)</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X862538218748627F">8.2 <span class="Heading">Poincare Series for Mod-<span class="SimpleMath">\(p\)</span> cohomology</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X780DF87680C3F52B">8.3 <span class="Heading">Functorial ring homomorphisms in Mod-<span class="SimpleMath">\(p\)</span> cohomology</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X834CED9D7A104695">8.3-1 <span class="Heading">Testing homomorphism properties</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7A0D505D844F0CD4">8.3-2 <span class="Heading">Testing functorial properties</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X855764877FA44225">8.3-3 <span class="Heading">Computing with larger groups</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X80114B0483EF9A67">8.4 <span class="Heading">Steenrod operations for finite <span class="SimpleMath">\(2\)</span>-groups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X7D5ACA56870A40E9">8.5 <span class="Heading">Steenrod operations on the classifying space of a finite <span class="SimpleMath">\(p\)</span>-group</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8_mj.html#X7D2D26C0784A0E14">8.6 <span class="Heading">Mod-<span class="SimpleMath">\(p\)</span> cohomology rings of crystallographic groups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X81C107C07CF02F0E">8.6-1 <span class="Heading">Poincare series for crystallographic groups</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8_mj.html#X7F5C242F7BC938A5">8.6-2 <span class="Heading">Mod <span class="SimpleMath">\(2\)</span> cohomology rings of <span class="SimpleMath">\(3\)</span>-dimensional crystallographic groups</span></a>
</span>
</div></div>
</div>
<h3>8 <span class="Heading">Cohomology rings and Steenrod operations for groups</span></h3>
<p><a id="X877CAF8B7E64DE04" name="X877CAF8B7E64DE04"></a></p>
<h4>8.1 <span class="Heading">Mod-<span class="SimpleMath">\(p\)</span> cohomology rings of finite groups</span></h4>
<p>For a finite group <span class="SimpleMath">\(G\)</span>, prime <span class="SimpleMath">\(p\)</span> and positive integer <span class="SimpleMath">\(deg\)</span> the function <code class="code">ModPCohomologyRing(G,p,deg)</code> computes a finite dimensional graded ring equal to the cohomology ring <span class="SimpleMath">\(H^{\le deg}(G,\mathbb Z_p) := H^\ast(G,\mathbb Z_p)/\{x=0\ :\ {\rm degree}(x)>deg \}\)</span> .</p>
<p>The following example computes the first <span class="SimpleMath">\(14\)</span> degrees of the cohomology ring <span class="SimpleMath">\(H^\ast(M_{11},\mathbb Z_2)\)</span> where <span class="SimpleMath">\(M_{11}\)</span> is the Mathieu group of order <span class="SimpleMath">\(7920\)</span>. The ring is seen to be generated by three elements <span class="SimpleMath">\(a_3, a_4, a_6\)</span> in degrees <span class="SimpleMath">\(3,4,5\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=MathieuGroup(11);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p:=2;;deg:=14;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=ModPCohomologyRing(G,p,deg);</span>
<algebra over GF(2), with 20 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">gns:=ModPRingGenerators(A);</span>
[ v.1, v.6, v.8+v.10, v.13 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(gns,A!.degree);</span>
[ 0, 3, 4, 5 ]
</pre></div>
<p>The following additional command produces a rational function <span class="SimpleMath">\(f(x)\)</span> whose series expansion <span class="SimpleMath">\(f(x) = \sum_{i=0}^\infty f_ix^i\)</span> has coefficients <span class="SimpleMath">\(f_i\)</span> which are guaranteed to satisfy <span class="SimpleMath">\(f_i = \dim H^i(G,\mathbb Z_p)\)</span> in the range <span class="SimpleMath">\(0\le i\le deg\)</span>. We refer to <span class="SimpleMath">\(f(x)\)</span> as the <em>Poincare series</em> for the group at the prime <span class="SimpleMath">\(p=2\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=PoincareSeries(A);</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)
<span class="GAPprompt">gap></span> <span class="GAPinput">Let's use f to list the first few cohomology dimensions</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ExpansionOfRationalFunction(f,deg); </span>
[ 1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2 ]
</pre></div>
<p>An alternative command for computing the Poincare series is the following. In this alternative we choose to ensure correctness in degrees <span class="SimpleMath">\(\le 100\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PoincareSeriesPrimePart(MathieuGroup(11),2,100);</span>
The series is guaranteed correct for group cohomology in degrees < 101
(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>If one needs to verify that the Poincare series is valid in all degrees then more work is required. One readily implemented (but computationally non-optimal) approach is to use Peter Symmonds result <a href="chapBib_mj.html#biBSymmonds">[Sym10]</a> that: if a non-cyclic finite group <span class="SimpleMath">\(G\)</span> has a faithful complex representation equal to a sum of irreducibles of dimensions <span class="SimpleMath">\(n_i\)</span> then the cohomology ring <span class="SimpleMath">\(H^\ast(G,\mathbb Z_p)\)</span> is generated by elements of degree at most <span class="SimpleMath">\(\sum n_i^2\)</span>; a degree bound for the relations is <span class="SimpleMath">\(2 \sum n_i^2\)</span>. The following commands use this bound, in conjunction with Webb's result <a href="chap7_mj.html#X7C5233E27D2D603E"><span class="RefLink">7.14</span></a> on the Quillen complex, to obtained a Poincare series that is guaranteed correct in all degree.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=MathieuGroup(11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:=HomologicalGroupDecomposition(G,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ModPCohomologyPresentationBounds(h[1][1]);</span>
rec( generators_degree_bound := 4, relators_degree_bound := 8 )
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=ModPCohomologyRing(h[1][1],2,9);;F:=PresentationOfGradedStructureConstantAlgebra(A);;f11:=HilbertPoincareSeries(F);</span>
(x_1^2-x_1+1)/(x_1^4-2*x_1^3+2*x_1^2-2*x_1+1)
<span class="GAPprompt">gap></span> <span class="GAPinput">ModPCohomologyPresentationBounds(h[1][2]);</span>
rec( generators_degree_bound := 9, relators_degree_bound := 18 )
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=ModPCohomologyRing(h[1][2],2,19);;F:=PresentationOfGradedStructureConstantAlgebra(A);;f12:=HilbertPoincareSeries(F);</span>
(x_1^2+1)/(x_1^4-x_1^3-x_1+1)
<span class="GAPprompt">gap></span> <span class="GAPinput">ModPCohomologyPresentationBounds(h[2][1]);</span>
rec( generators_degree_bound := 4, relators_degree_bound := 8 )
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=ModPCohomologyRing(h[2][1],2,9);;F:=PresentationOfGradedStructureConstantAlgebra(A);;f21:=HilbertPoincareSeries(F);</span>
(1)/(x_1^2-2*x_1+1)
<span class="GAPprompt">gap></span> <span class="GAPinput">f11+f12-f21;</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="X870E0299782638AF" name="X870E0299782638AF"></a></p>
<h5>8.1-1 <span class="Heading">Ring presentations (for the commutative <span class="SimpleMath">\(p=2\)</span> case)</span></h5>
<p>The cohomology ring <span class="SimpleMath">\(H^\ast(G,\mathbb Z_p)\)</span> is graded commutative which, in the case <span class="SimpleMath">\(p=2\)</span>, implies strictly commutative. The following additional commands can be applied in the <span class="SimpleMath">\(p=2\)</span> setting to determine a presentation for a graded commutative ring <span class="SimpleMath">\(F\)</span> that is guaranteed to be isomorphic to the cohomology ring <span class="SimpleMath">\(H^\ast(G,\mathbb Z_p)\)</span> in degrees <span class="SimpleMath">\(i\le deg\)</span>. If <span class="SimpleMath">\(deg\)</span> is chosen "sufficiently large" then <span class="SimpleMath">\(F\)</span> will be isomorphic to the cohomology ring.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=PresentationOfGradedStructureConstantAlgebra(A);</span>
Graded algebra GF(2)[ x_1, x_2, x_3 ] / [ x_1^2*x_2+x_3^2
] with indeterminate degrees [ 3, 4, 5 ]
</pre></div>
<p>The additional command</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">p:=HilbertPoincareSeries(F);</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>invokes a call to <strong class="button">Singular</strong> in order to calculate the Poincare series of the graded algebra <span class="SimpleMath">\(F\)</span>.</p>
<p><a id="X862538218748627F" name="X862538218748627F"></a></p>
<h4>8.2 <span class="Heading">Poincare Series for Mod-<span class="SimpleMath">\(p\)</span> cohomology</span></h4>
<p>For a finite <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> the command <code class="code">PoincarePolynomial(G)</code> returns a rational function <span class="SimpleMath">\(f(x)=p(x)/q(x)\)</span> whose series expansion <span class="SimpleMath">\(f(x) = \sum_{i=0}^\infty f_ix^i\)</span> has coefficients <span class="SimpleMath">\(f_i\)</span> that are guaranteed to satisfy <span class="SimpleMath">\(f_i = \dim H^i(G,\mathbb Z_p)\)</span> in the range <span class="SimpleMath">\(0\le i < 1+ deg\)</span> for some displayed value of <span class="SimpleMath">\(deg\)</span>. Furthermore, the coefficients <span class="SimpleMath">\(f_i\)</span> are guaranteed to be integers for all <span class="SimpleMath">\(0\le i\le 1000\)</span> and the order of the pole of <span class="SimpleMath">\(f(x)\)</span> at <span class="SimpleMath">\(x=1\)</span> is guaranteed to equal the <span class="SimpleMath">\(p\)</span>-rank of <span class="SimpleMath">\(G\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SmallGroup(3^4,10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(G);</span>
"C3 . ((C3 x C3) : C3) = (C3 x C3) . (C3 x C3)"
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=PoincareSeries(G);</span>
The series is guaranteed correct for group cohomology in degrees < 14
(-x_1^3+x_1^2+1)/(x_1^6-2*x_1^5+2*x_1^4-2*x_1^3+2*x_1^2-2*x_1+1)
</pre></div>
<p>If a higher value of <span class="SimpleMath">\(deg\)</span> is required then this can be entered as an optional second argument. For instance, the following increases the value to <span class="SimpleMath">\(deg=100\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=PoincareSeries(G,100);</span>
The series is guaranteed correct for group cohomology in degrees < 101
(-x_1^3+x_1^2+1)/(x_1^6-2*x_1^5+2*x_1^4-2*x_1^3+2*x_1^2-2*x_1+1)
</pre></div>
<p>As mentioned above, one approach to verifying that the Poincare series is valid in all degrees is to use Peter Symmonds result <a href="chapBib_mj.html#biBSymmonds">[Sym10]</a> that: if a non-cyclic finite group <span class="SimpleMath">\(G\)</span> has a faithful complex representation equal to a sum of irreducibles of dimensions <span class="SimpleMath">\(n_i\)</span> then the cohomology ring <span class="SimpleMath">\(H^\ast(G,\mathbb Z_p)\)</span> is generated by elements of degree at most <span class="SimpleMath">\(\sum n_i^2\)</span>; a degree bound for the relations is <span class="SimpleMath">\(2 \sum n_i^2\)</span>. Thus, if we use at least <span class="SimpleMath">\(\sum n_i^2\)</span> degrees of a resolution to construct a presentation for the cohomology ring then the presented ring maps surjectively onto the actual cohomology ring. Furthermore, if this surjection is a bijection in the first <span class="SimpleMath">\(2 \sum n_i^2\)</span> degrees then it is necessarily an isomorphism in all degrees.</p>
<p>The following commands use this approach to obtain a guaranteed presentation and Poincare series for the Sylow <span class="SimpleMath">\(2\)</span>-subgroup of the Mathieu group <span class="SimpleMath">\(M_{12}\)</span>.</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">ModPCohomologyPresentationBounds(G);</span>
rec( generators_degree_bound := 16, relators_degree_bound := 32 )
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=ModPCohomologyRing(G,16);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=PresentationOfGradedStructureConstantAlgebra(A);</span>
Graded algebra GF(2)[ x_1, x_2, x_3, x_4, x_5, x_6, x_7 ] /
[ x_1*x_3, x_1*x_2, x_1*x_4, x_2*x_3^2+x_3^3+x_3*x_4+x_3*x_5,
x_2*x_6+x_3*x_6+x_4*x_5, x_2*x_3*x_4+x_3^2*x_4+x_3*x_6,
x_2^2*x_4+x_3^2*x_4+x_3*x_6+x_4^2, x_3^2*x_6+x_3*x_4^2+x_3*x_4*x_5,
x_2*x_4*x_5+x_4*x_6, x_3^2*x_4*x_5+x_3*x_4*x_6+x_3*x_5*x_6,
x_1^3*x_6+x_1^2*x_7+x_1*x_5*x_6+x_3*x_5*x_6+x_4*x_5^2+x_6^2,
x_3*x_4^2*x_5+x_3*x_6^2 ] with indeterminate degrees [ 1, 1, 1, 2, 2, 3, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=HilbertPoincareSeries(F);</span>
(1)/(-x_1^3+3*x_1^2-3*x_1+1)
<span class="GAPprompt">gap></span> <span class="GAPinput">ff:=PoincareSeries(G,32);</span>
The series is guaranteed correct for group cohomology in degrees < 33
(1)/(-x_1^3+3*x_1^2-3*x_1+1)
</pre></div>
<p>An alternative approach to obtaining a guaranteed presentation is to implement Len even's spectral sequence proof of the finite generation of cohomology rings of finite groups. The following example determines a guaranteed presentation in this way for the cohomology ring <span class="SimpleMath">\(H^\ast(Syl_2(M_{12}),\mathbb Z_2)\)</span>. The Lyndon-Hochschild-Serre spectral sequence, and Groebner basis routines from <strong class="button">Singular</strong> (for commutative rings), are used to determine how much of a resolution is needed to compute the guaranteed correct presentation.</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">F:=Mod2CohomologyRingPresentation(G);</span>
Alpha version of completion test code will be used. This needs further work.
Graded algebra GF(2)[ x_1, x_2, x_3, x_4, x_5, x_6, x_7 ] /
[ x_2*x_3, x_1*x_2, x_2*x_4, x_1^3+x_1^2*x_3+x_1*x_5,
x_1*x_3*x_4+x_1*x_3*x_5+x_3^2*x_4+x_1*x_6+x_3*x_6+x_4*x_5,
x_1^2*x_4+x_1^2*x_5+x_1*x_3*x_5+x_3^2*x_4+x_1*x_6+x_4^2,
x_1^2*x_3^2+x_1^2*x_5+x_1*x_3*x_5+x_1*x_6+x_3*x_6+x_4^2+x_4*x_5,
x_1^2*x_6+x_1*x_3*x_6+x_1*x_4*x_5+x_3^2*x_6+x_3*x_4^2+x_3*x_4*x_5,
x_1*x_3^2*x_5+x_3^3*x_4+x_1*x_3*x_6+x_1*x_4^2+x_3^2*x_6+x_3*x_4^2+x_4*x_6,
x_1^2*x_3*x_5+x_1*x_3*x_6+x_1*x_4^2+x_1*x_5^2,
x_3^3*x_6+x_3^2*x_4^2+x_3^2*x_4*x_5+x_3*x_4*x_6+x_3*x_5*x_6+x_4^3+x_4*x_5^2,
x_1*x_3^2*x_6+x_1*x_4*x_6+x_2^2*x_7+x_2*x_5*x_6+x_3*x_4*x_6+x_3*x_5*x_6+x_6^2,
x_1^2*x_5^2+x_1*x_3*x_5^2+x_3^2*x_4^2+x_3^2*x_4*x_5+x_2^2*x_7+x_2*x_5*x_6+x_3*x_5*x_6+x_6^2 ]
with indeterminate degrees [ 1, 1, 1, 2, 2, 3, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=HilbertPoincareSeries(F);</span>
(1)/(-x_1^3+3*x_1^2-3*x_1+1)
</pre></div>
<p><a id="X780DF87680C3F52B" name="X780DF87680C3F52B"></a></p>
<h4>8.3 <span class="Heading">Functorial ring homomorphisms in Mod-<span class="SimpleMath">\(p\)</span> cohomology</span></h4>
<p>The following example constructs the ring homomorphism</p>
<p><span class="SimpleMath">\(F\colon H^{\le deg}(G,\mathbb Z_p) \rightarrow H^{\le deg}(H,\mathbb Z_p)\)</span></p>
<p>induced by the group homomorphism <span class="SimpleMath">\(f\colon H\rightarrow G\)</span> with <span class="SimpleMath">\(H=A_5\)</span>, <span class="SimpleMath">\(G=S_5\)</span>, <span class="SimpleMath">\(f\)</span> the canonical inclusion of the alternating group into the symmetric group, <span class="SimpleMath">\(p=2\)</span> and <span class="SimpleMath">\(deg=7\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SymmetricGroup(5);;H:=AlternatingGroup(5);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=GroupHomomorphismByFunction(H,G,x->x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">p:=2;; deg:=7;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=ModPCohomologyRing(f,p,deg);</span>
[ v.1, v.2, v.4+v.6, v.5, v.7, v.8, v.9, v.12+v.15, v.13, v.14, v.16+v.17,
v.18, v.19, v.20, v.22+v.24+v.28, v.23, v.25, v.26, v.27 ] ->
[ v.1, 0*v.1, v.4+v.5+v.6, 0*v.1, v.7+v.8, 0*v.1, 0*v.1, v.14+v.15, 0*v.1,
0*v.1, v.16+v.17+v.19, 0*v.1, 0*v.1, 0*v.1, v.22+v.23+v.26+v.27+v.28,
v.25, 0*v.1, 0*v.1, 0*v.1 ]
</pre></div>
<p><a id="X834CED9D7A104695" name="X834CED9D7A104695"></a></p>
<h5>8.3-1 <span class="Heading">Testing homomorphism properties</span></h5>
<p>The following commands are consistent with <span class="SimpleMath">\(F\)</span> being a ring homomorphism.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x:=Random(Source(F));</span>
v.4+v.6+v.8+v.9+v.12+v.13+v.14+v.15+v.18+v.20+v.22+v.24+v.25+v.28+v.32+v.35
<span class="GAPprompt">gap></span> <span class="GAPinput">y:=Random(Source(F));</span>
v.1+v.2+v.7+v.9+v.13+v.23+v.26+v.27+v.32+v.33+v.34+v.35
<span class="GAPprompt">gap></span> <span class="GAPinput">Image(F,x)+Image(F,y)=Image(F,x+y);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Image(F,x)*Image(F,y)=Image(F,x*y);</span>
true
</pre></div>
<p><a id="X7A0D505D844F0CD4" name="X7A0D505D844F0CD4"></a></p>
<h5>8.3-2 <span class="Heading">Testing functorial properties</span></h5>
<p>The following example takes two "random" automorphisms <span class="SimpleMath">\(f,g\colon K\rightarrow K\)</span> of the group <span class="SimpleMath">\(K\)</span> of order <span class="SimpleMath">\(24\)</span> arising as the direct product <span class="SimpleMath">\(K=C_3\times Q_8\)</span> and constructs the three ring isomorphisms <span class="SimpleMath">\(F,G,FG\colon H^{\le 5}(K,\mathbb Z_2) \rightarrow H^{\le 5}(K,\mathbb Z_2)\)</span> induced by <span class="SimpleMath">\(f, g\)</span> and the composite <span class="SimpleMath">\(f\circ g\)</span>. It tests that <span class="SimpleMath">\(FG\)</span> is indeed the composite <span class="SimpleMath">\(G\circ F\)</span>. Note that when we create the ring <span class="SimpleMath">\(H^{\le 5}(K,\mathbb Z_2)\)</span> twice in <strong class="button">GAP</strong> we obtain two canonically isomorphic but distinct implimentations of the ring. Thus the canocial isomorphism between these distinct implementations needs to be incorporated into the test. Note also that <strong class="button">GAP</strong> defines <span class="SimpleMath">\(g\ast f = f\circ g\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=SmallGroup(24,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">aut:=AutomorphismGroup(K);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=Elements(aut)[5];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Elements(aut)[8];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">fg:=g*f;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=ModPCohomologyRing(f,2,5);</span>
[ v.1, v.2, v.3, v.4, v.5, v.6, v.7 ] -> [ v.1, v.2+v.3, v.3, v.4+v.5, v.5,
v.6, v.7 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=ModPCohomologyRing(g,2,5);</span>
[ v.1, v.2, v.3, v.4, v.5, v.6, v.7 ] -> [ v.1, v.2+v.3, v.2, v.5, v.4+v.5,
v.6, v.7 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">FG:=ModPCohomologyRing(fg,2,5);</span>
[ v.1, v.2, v.3, v.4, v.5, v.6, v.7 ] -> [ v.1, v.3, v.2, v.4, v.4+v.5, v.6,
v.7 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">sF:=Source(F);;tF:=Target(F);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sG:=Source(G);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">tGsF:=AlgebraHomomorphismByImages(tF,sG,Basis(tF),Basis(sG));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(GeneratorsOfAlgebra(sF),x->Image(G,Image(tGsF,Image(F,x))));</span>
[ v.1, v.3, v.2, v.4, v.4+v.5, v.6, v.7 ]
</pre></div>
<p><a id="X855764877FA44225" name="X855764877FA44225"></a></p>
<h5>8.3-3 <span class="Heading">Computing with larger groups</span></h5>
<p>Mod-<span class="SimpleMath">\(p\)</span> cohomology rings of finite groups are constructed as the rings of stable elements in the cohomology of a (non-functorially) chosen Sylow <span class="SimpleMath">\(p\)</span>-subgroup and thus require the construction of a free resolution only for the Sylow subgroup. However, to ensure the functoriality of induced cohomology homomorphisms the above computations construct free resolutions for the entire groups <span class="SimpleMath">\(G,H\)</span>. This is a more expensive computation than finding resolutions just for Sylow subgroups.</p>
<p>The default algorithm used by the function <code class="code">ModPCohomologyRing()</code> for constructing resolutions of a finite group <span class="SimpleMath">\(G\)</span> is <code class="code">ResolutionFiniteGroup()</code> or <code class="code">ResolutionPrimePowerGroup()</code> in the case when <span class="SimpleMath">\(G\)</span> happens to be a group of prime-power order. If the user is able to construct the first <span class="SimpleMath">\(deg\)</span> terms of free resolutions <span class="SimpleMath">\(RG, RH\)</span> for the groups <span class="SimpleMath">\(G, H\)</span> then the pair <code class="code">[RG,RH]</code> can be entered as the third input variable of <code class="code">ModPCohomologyRing()</code>.</p>
<p>For instance, the following example constructs the ring homomorphism</p>
<p><span class="SimpleMath">\(F\colon H^{\le 7}(A_6,\mathbb Z_2) \rightarrow H^{\le 7}(S_6,\mathbb Z_2)\)</span></p>
<p>induced by the the canonical inclusion of the alternating group <span class="SimpleMath">\(A_6\)</span> into the symmetric group <span class="SimpleMath">\(S_6\)</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">H:=AlternatingGroup(6);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:=GroupHomomorphismByFunction(H,G,x->x);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RG:=ResolutionFiniteGroup(G,7);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">RH:=ResolutionFiniteSubgroup(RG,H);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:=ModPCohomologyRing(f,2,[RG,RH]); </span>
[ v.1, v.2+v.3, v.6+v.8+v.10, v.7+v.9, v.11+v.12, v.13+v.15+v.16+v.18+v.19,
v.14+v.16+v.19, v.17, v.22, v.23+v.28+v.32+v.35,
v.24+v.26+v.27+v.29+v.32+v.33+v.35, v.25+v.26+v.27+v.29+v.32+v.33+v.35,
v.30+v.32+v.33+v.34+v.35, v.36+v.39+v.43+v.45+v.47+v.49+v.50+v.55,
v.38+v.45+v.47+v.49+v.50+v.55, v.40,
v.41+v.43+v.45+v.47+v.48+v.49+v.50+v.53+v.55,
v.42+v.43+v.45+v.46+v.47+v.49+v.53+v.54, v.44+v.45+v.46+v.47+v.49+v.53+v.54,
v.51+v.52, v.58+v.60, v.59+v.68+v.73+v.77+v.81+v.83,
v.62+v.68+v.74+v.77+v.78+v.80+v.81+v.83+v.84,
v.63+v.69+v.73+v.74+v.78+v.80+v.84, v.64+v.68+v.73+v.77+v.81+v.83, v.65,
v.66+v.75+v.81, v.67+v.68+v.69+v.70+v.73+v.74+v.78+v.80+v.84,
v.71+v.72+v.73+v.76+v.77+v.78+v.80+v.82+v.83+v.84, v.79 ] ->
[ v.1, 0*v.1, v.4+v.5+v.6, 0*v.1, v.8, v.8, 0*v.1, v.7, 0*v.1,
v.12+v.13+v.14+v.15, v.12+v.13+v.14+v.15, v.12+v.13+v.14+v.15,
v.12+v.13+v.14+v.15, v.18+v.19, 0*v.1, 0*v.1, v.18+v.19, v.18+v.19,
v.18+v.19, v.16+v.17, 0*v.1, v.25, v.22+v.24+v.25+v.26+v.27+v.28,
v.22+v.24+v.25+v.26+v.27+v.28, 0*v.1, 0*v.1, v.25, v.22+v.24+v.26+v.27+v.28,
v.22+v.24+v.26+v.27+v.28, v.23 ]
</pre></div>
<p><a id="X80114B0483EF9A67" name="X80114B0483EF9A67"></a></p>
<h4>8.4 <span class="Heading">Steenrod operations for finite <span class="SimpleMath">\(2\)</span>-groups</span></h4>
<p>The command <code class="code">CohomologicalData(G,n)</code> prints complete information for the cohomology ring <span class="SimpleMath">\(H^\ast(G, Z_2 )\)</span> and steenrod operations for a <span class="SimpleMath">\(2\)</span>-group <span class="SimpleMath">\(G\)</span> provided that the integer <span class="SimpleMath">\(n\)</span> is at least the maximal degree of a generator or relator in a minimal set of generatoirs and relators for the ring.</p>
<p>The following example produces complete information on the Steenrod algebra of group number <span class="SimpleMath">\(8\)</span> in <strong class="button">GAP</strong>'s library of groups of order <span class="SimpleMath">\(32\)</span>. Groebner basis routines (for commutative rings) from <strong class="button">Singular</strong> are called in the example. (This example take over 2 hours to run. Most other groups of order 32 run significantly quicker.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">CohomologicalData(SmallGroup(32,8),12);</span>
Integer argument is large enough to ensure completeness of cohomology ring presentation.
Group number: 8
Group description: C2 . ((C4 x C2) : C2) = (C2 x C2) . (C4 x C2)
Cohomology generators
Degree 1: a, b
Degree 2: c, d
Degree 3: e
Degree 5: f, g
Degree 6: h
Degree 8: p
Cohomology relations
1: f^2
2: c*h+e*f
3: c*f
4: b*h+c*g
5: b*e+c*d
6: a*h
7: a*g
8: a*f+b*f
9: a*e+c^2
10: a*c
11: a*b
12: a^2
13: d*e*h+e^2*g+f*h
14: d^2*h+d*e*f+d*e*g+f*g
15: c^2*d+b*f
16: b*c*g+e*f
17: b*c*d+c*e
18: b^2*g+d*f
19: b^2*c+c^2
20: b^3+a*d
21: c*d^2*e+c*d*g+d^2*f+e*h
22: c*d^3+d*e^2+d*h+e*f+e*g
23: b^2*d^2+c*d^2+b*f+e^2
24: b^3*d
25: d^3*e^2+d^2*e*f+c^2*p+h^2
26: d^4*e+b*c*p+e^2*g+g*h
27: d^5+b*d^2*g+b^2*p+f*g+g^2
Poincare series
(x^5+x^2+1)/(x^8-2*x^7+2*x^6-2*x^5+2*x^4-2*x^3+2*x^2-2*x+1)
Steenrod squares
Sq^1(c)=0
Sq^1(d)=b*b*b+d*b
Sq^1(e)=c*b*b
Sq^2(e)=e*d+f
Sq^1(f)=c*d*b*b+d*d*b*b
Sq^2(f)=g*b*b
Sq^4(f)=p*a
Sq^1(g)=d*d*d+g*b
Sq^2(g)=0
Sq^4(g)=c*d*d*d*b+g*d*b*b+g*d*d+p*a+p*b
Sq^1(h)=c*d*d*b+e*d*d
Sq^2(h)=d*d*d*b*b+c*d*d*d+g*c*b
Sq^4(h)=d*d*d*d*b*b+g*e*d+p*c
Sq^1(p)=c*d*d*d*b
Sq^2(p)=d*d*d*d*b*b+c*d*d*d*d
Sq^4(p)=d*d*d*d*d*b*b+d*d*d*d*d*d+g*d*d*d*b+g*g*d+p*d*d
</pre></div>
<p><a id="X7D5ACA56870A40E9" name="X7D5ACA56870A40E9"></a></p>
<h4>8.5 <span class="Heading">Steenrod operations on the classifying space of a finite <span class="SimpleMath">\(p\)</span>-group</span></h4>
<p>The following example constructs the first eight degrees of the mod-<span class="SimpleMath">\(3\)</span> cohomology ring <span class="SimpleMath">\(H^\ast(G,\mathbb Z_3)\)</span> for the group <span class="SimpleMath">\(G\)</span> number 4 in <strong class="button">GAP</strong>'s library of groups of order <span class="SimpleMath">\(81\)</span>. It determines a minimal set of ring generators lying in degree <span class="SimpleMath">\(\le 8\)</span> and it evaluates the Bockstein operator on these generators. Steenrod powers for <span class="SimpleMath">\(p\ge 3\)</span> are not implemented as no efficient method of implementation is known.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SmallGroup(81,4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:=ModPSteenrodAlgebra(G,8);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(ModPRingGenerators(A),x->Bockstein(A,x));</span>
[ 0*v.1, 0*v.1, v.5, 0*v.1, (Z(3))*v.7+v.8+(Z(3))*v.9 ]
</pre></div>
<p><a id="X7D2D26C0784A0E14" name="X7D2D26C0784A0E14"></a></p>
<h4>8.6 <span class="Heading">Mod-<span class="SimpleMath">\(p\)</span> cohomology rings of crystallographic groups</span></h4>
<p>Mod <span class="SimpleMath">\(p\)</span> cohomology ring computations can be attempted for any group <span class="SimpleMath">\(G\)</span> for which we can compute sufficiently many terms of a free <span class="SimpleMath">\(ZG\)</span>-resolution with contracting homotopy. The contracting homotopy is not needed if only the dimensions of the cohomology in each degree are sought. Crystallographic groups are one class of infinite groups where such computations can be attempted.</p>
<p><a id="X81C107C07CF02F0E" name="X81C107C07CF02F0E"></a></p>
<h5>8.6-1 <span class="Heading">Poincare series for crystallographic groups</span></h5>
<p>Consider the space group <span class="SimpleMath">\(G=SpaceGroupOnRightIT(3,226,'1')\)</span>. The following computation computes the infinite series</p>
<p><span class="SimpleMath">\((-2x^4+2x^2+1)/(-x^5+2x^4-x^3+x^2-2x+1)\)</span></p>
<p>in which the coefficient of the monomial <span class="SimpleMath">\(x^n\)</span> is guaranteed to equal the dimension of the vector space <span class="SimpleMath">\(H^n(G,\mathbb Z_2)\)</span> in degrees <span class="SimpleMath">\(n\le 14\)</span>. One would need to involve a theoretical argument to establish that this equality in fact holds in every degree <span class="SimpleMath">\(n\ge 0\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SpaceGroupIT(3,226);</span>
SpaceGroupOnRightIT(3,226,'1')
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionSpaceGroup(G,15);</span>
Resolution of length 15 in characteristic 0 for <matrix group with
8 generators> .
No contracting homotopy available.
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=List([0..14],n->Cohomology(HomToIntegersModP(R,2),n));</span>
[ 1, 2, 5, 9, 11, 15, 20, 23, 28, 34, 38, 44, 51, 56, 63 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">PoincareSeries(D,14);</span>
(-2*x_1^4+2*x_1^2+1)/(-x_1^5+2*x_1^4-x_1^3+x_1^2-2*x_1+1)
</pre></div>
<p>Consider the space group <span class="SimpleMath">\(SpaceGroupOnRightIT(3,103,'1')\)</span>. The following computation uses a different construction of a free resolution to compute the infinite series</p>
<p><span class="SimpleMath">\( (x^3+2x^2+2x+1)/(-x+1) \)</span></p>
<p>in which the coefficient of the monomial <span class="SimpleMath">\(x^n\)</span> is guaranteed to equal the dimension of the vector space <span class="SimpleMath">\(H^n(G,\mathbb Z_2)\)</span> in degrees <span class="SimpleMath">\(n\le 99\)</span>. The final commands show that <span class="SimpleMath">\(G\)</span> acts on a (cubical) cellular decomposition of <span class="SimpleMath">\(\mathbb R^3\)</span> with cell ctabilizers being either trivial or cyclic of order <span class="SimpleMath">\(2\)</span> or <span class="SimpleMath">\(4\)</span>. From this extra calculation it follows that the cohomology is periodic in degrees greater than <span class="SimpleMath">\(3\)</span> and that the Poincare series is correct in every degree <span class="SimpleMath">\(n \ge 0\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SpaceGroupIT(3,103);</span>
SpaceGroupOnRightIT(3,103,'1')
<span class="GAPprompt">gap></span> <span class="GAPinput">R:=ResolutionCubicalCrystGroup(G,100);</span>
Resolution of length 100 in characteristic 0 for <matrix group with 6 generators> .
<span class="GAPprompt">gap></span> <span class="GAPinput">D:=List([0..99],n->Cohomology(HomToIntegersModP(R,2),n));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">PoincareSeries(D,99);</span>
(x_1^3+2*x_1^2+2*x_1+1)/(-x_1+1)
#Torsion subgroups are cyclic
<span class="GAPprompt">gap></span> <span class="GAPinput">B:=CrystGFullBasis(G);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">C:=CrystGcomplex(GeneratorsOfGroup(G),B,1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">for n in [0..3] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">for k in [1..C!.dimension(n)] do</span>
<span class="GAPprompt">></span> <span class="GAPinput">Print(StructureDescription(C!.stabilizer(n,k))," ");</span>
<span class="GAPprompt">></span> <span class="GAPinput">od;od;</span>
C4 C2 C4 1 1 C4 C2 C4 1 1 1 1
</pre></div>
<p><a id="X7F5C242F7BC938A5" name="X7F5C242F7BC938A5"></a></p>
<h5>8.6-2 <span class="Heading">Mod <span class="SimpleMath">\(2\)</span> cohomology rings of <span class="SimpleMath">\(3\)</span>-dimensional crystallographic groups</span></h5>
<p>Computations in the <em>integral</em> cohomology of a crystallographic group are illustrated in Section <a href="chap1_mj.html#X86881717878ADCD6"><span class="RefLink">1.19</span></a>. The commands underlying that illustration could be further developed and adapted to mod <span class="SimpleMath">\(p\)</span> cohomology. Indeed, the authors of the paper <a href="chapBib_mj.html#biBliuye">[LY24a]</a> have developed commands for accessing the mod <span class="SimpleMath">\(2\)</span> cohomology of <span class="SimpleMath">\(3\)</span>-dimensional crystallographic groups with the aim of establishing a connection between these rings and the lattice structure of crystals with space group symmetry. Their code is available at the github repository <a href="chapBib_mj.html#biBliuyegithub">[LY24b]</a>. In particular, their code contains the command</p>
<ul>
<li><p><code class="code">SpaceGroupCohomologyRingGapInterface(ITC)</code></p>
</li>
</ul>
<p>that inputs an integer in the range <span class="SimpleMath">\(1\le ITC\le 230\)</span> corresponding to the numbering of a <span class="SimpleMath">\(3\)</span>-dimensional space group <span class="SimpleMath">\(G\)</span> in the International Table for Crystallography. This command returns</p>
<ul>
<li><p>a presentation for the mod <span class="SimpleMath">\(2\)</span> cohomology ring <span class="SimpleMath">\(H^\ast(G,\mathbb Z_2)\)</span>. The presentation is guaranteed to be correct for low degree cohomology. In cases where the cohomology is periodic in degrees <span class="SimpleMath">\( \ge 5\)</span> (which can be tested using <code class="code">IsPeriodicSpaceGroup(G)</code>) the presentation is guaranteed correct in all degrees. In non-periodic cases some additional mathematical argument needs to be provided to be mathematically sure that the presentation is correct in all degrees.</p>
</li>
<li><p>the Lieb-Schultz-Mattis anomaly (degree-3 cocycles) associated with the Irreducible Wyckoff Position (see the paper <a href="chapBib_mj.html#biBliuye">[LY24a]</a> for a definition).</p>
</li>
</ul>
<p>The command <code class="code">SpaceGroupCohomologyRingGapInterface(ITC)</code> is fast for most groups (a few seconds to a few minutes) but can be very slow for certain space groups (e.g. ITC <span class="SimpleMath">\(= 228\)</span> and ITC <span class="SimpleMath">\(= 142\)</span>). The following illustration assumes that two relevant files have been downloaded from <a href="chapBib_mj.html#biBliuyegithub">[LY24b]</a> and illustrates the command for ITC <span class="SimpleMath">\( =30\)</span> and ITC <span class="SimpleMath">\(=216\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Read("SpaceGroupCohomologyData.gi"); #These two files must be </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Read("SpaceGroupCohomologyFunctions.gi"); #downloaded from</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"> #https://github.com/liuchx1993/Space-Group-Cohomology-and-LSM/</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPeriodicSpaceGroup(SpaceGroupIT(3,30));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">SpaceGroupCohomologyRingGapInterface(30);</span>
===========================================
Mod-2 Cohomology Ring of Group No. 30:
Z2[Ac,Am,Ax,Bb]/<R2,R3,R4>
R2: Ac.Am Am^2 Ax^2+Ac.Ax
R3: Am.Bb
R4: Bb^2
===========================================
LSM:
2a Ac.Bb+Ax.Bb
2b Ax.Bb
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPeriodicSpaceGroup(SpaceGroupIT(3,216));</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">SpaceGroupCohomologyRingGapInterface(216);</span>
===========================================
Mod-2 Cohomology Ring of Group No. 216:
Z2[Am,Ba,Bb,Bxyxzyz,Ca,Cb,Cc,Cxyz]/<R4,R5,R6>
R4: Am.Ca Am.Cb Ba.Bxyxzyz+Am.Cc Bb^2+Am.Cc+Ba.Bb Bb.Bxyxzyz+Am^2.Bb+Am.Cxyz Bxyxzyz^2
R5: Bxyxzyz.Ca Ba.Cb+Bb.Ca Bb.Cb+Bb.Ca Bxyxzyz.Cb Bxyxzyz.Cc Ba.Cxyz+Am.Ba.Bb+Bb.Cc Bb.Cxyz+Am^2.Cc+Am.Ba.Bb+Bb.Cc Bxyxzyz.Cxyz+Am^3.Bb+Am^2.Cxyz
===========================================
LSM:
4a Ca+Cc+Cxyz
4b Cb+Cc+Cxyz
4c Cb+Cxyz
4d Cxyz
true
</pre></div>
<p>In the example the naming convention for ring generators follows the paper <a href="chapBib_mj.html#biBliuye">[LY24a]</a>.</p>
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