GAP (HAP commands) - Chapter 8: Cohomology rings and Steenrod operations for groups

For a finite group \(G\), prime \(p\) and positive integer \(deg\) the function ModPCohomologyRing(G,p,deg) computes a finite dimensional graded ring equal to the cohomology ring \(H^{\le deg}(G,\mathbb Z_p) := H^\ast(G,\mathbb Z_p)/\{x=0\ :\ {\rm degree}(x)>deg \}\) .

The following example computes the first \(14\) degrees of the cohomology ring \(H^\ast(M_{11},\mathbb Z_2)\) where \(M_{11}\) is the Mathieu group of order \(7920\). The ring is seen to be generated by three elements \(a_3, a_4, a_6\) in degrees \(3,4,5\).

gap> G:=MathieuGroup(11);;          
gap> p:=2;;deg:=14;;
gap> A:=ModPCohomologyRing(G,p,deg);
<algebra over GF(2), with 20 generators>

gap> gns:=ModPRingGenerators(A);
[ v.1, v.6, v.8+v.10, v.13 ]
gap> List(gns,A!.degree);
[ 0, 3, 4, 5 ]

The following additional command produces a rational function \(f(x)\) whose series expansion \(f(x) = \sum_{i=0}^\infty f_ix^i\) has coefficients \(f_i\) which are guaranteed to satisfy \(f_i = \dim H^i(G,\mathbb Z_p)\) in the range \(0\le i\le deg\).

gap> f:=PoincareSeries(A);
(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)


gap> Let's use f to list the first few cohomology dimensions
gap> ExpansionOfRationalFunction(f,deg); 
[ 1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2 ]

8.1-1 Ring presentations (for the commutative \(p=2\) case)

The cohomology ring \(H^\ast(G,\mathbb Z_p)\) is graded commutative which, in the case \(p=2\), implies strictly commutative. The following additional commands can be applied in the \(p=2\) setting to determine a presentation for a graded commutative ring \(F\) that is guaranteed to be isomorphic to the cohomology ring \(H^\ast(G,\mathbb Z_p)\) in degrees \(i\le deg\). If \(deg\) is chosen "sufficiently large" then \(F\) will be isomorphic to the cohomology ring.

gap> F:=PresentationOfGradedStructureConstantAlgebra(A);
Graded algebra GF(2)[ x_1, x_2, x_3 ] / [ x_1^2*x_2+x_3^2 
 ] with indeterminate degrees [ 3, 4, 5 ]

gap> p:=HilbertPoincareSeries(F);
(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)

invokes a call to Singular in order to calculate the Poincare series of the graded algebra \(F\).

8.2 Functorial ring homomorphisms in Mod-\(p\) cohomology

induced by the group homomorphism \(f\colon H\rightarrow G\) with \(H=A_5\), \(G=S_5\), \(f\) the canonical inclusion of the alternating group into the symmetric group, \(p=2\) and \(deg=7\).

gap> G:=SymmetricGroup(5);;H:=AlternatingGroup(5);;
gap> f:=GroupHomomorphismByFunction(H,G,x->x);;
gap> p:=2;; deg:=7;;
gap> F:=ModPCohomologyRing(f,p,deg);
[ 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 ]

8.2-1 Testing homomorphism properties

gap> x:=Random(Source(F));
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
gap> y:=Random(Source(F));
v.1+v.2+v.7+v.9+v.13+v.23+v.26+v.27+v.32+v.33+v.34+v.35
gap> Image(F,x)+Image(F,y)=Image(F,x+y);
true
gap> Image(F,x)*Image(F,y)=Image(F,x*y);
true

8.2-2 Testing functorial properties

The following example takes two "random" automorphisms \(f,g\colon K\rightarrow K\) of the group \(K\) of order \(24\) arising as the direct product \(K=C_3\times Q_8\) and constructs the three ring isomorphisms \(F,G,FG\colon H^{\le 5}(K,\mathbb Z_2) \rightarrow H^{\le 5}(K,\mathbb Z_2)\) induced by \(f, g\) and the composite \(f\circ g\). It tests that \(FG\) is indeed the composite \(G\circ F\). Note that when we create the ring \(H^{\le 5}(K,\mathbb Z_2)\) twice in GAP 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 GAP defines \(g\ast f = f\circ g\).

gap> K:=SmallGroup(24,11);;
gap> aut:=AutomorphismGroup(K);;
gap> f:=Elements(aut)[5];;
gap> g:=Elements(aut)[8];;
gap> fg:=g*f;;
gap> F:=ModPCohomologyRing(f,2,5);
[ 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 ]
gap> G:=ModPCohomologyRing(g,2,5);
[ 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 ]
gap> FG:=ModPCohomologyRing(fg,2,5);
[ 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 ]

gap> sF:=Source(F);;tF:=Target(F);;
gap> sG:=Source(G);; 
gap> tGsF:=AlgebraHomomorphismByImages(tF,sG,Basis(tF),Basis(sG));;
gap> List(GeneratorsOfAlgebra(sF),x->Image(G,Image(tGsF,Image(F,x))));
[ v.1, v.3, v.2, v.4, v.4+v.5, v.6, v.7 ]

8.2-3 Computing with larger groups

Mod-\(p\) cohomology rings of finite groups are constructed as the rings of stable elements in the cohomology of a (non-functorially) chosen Sylow \(p\)-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 \(G,H\). This is a more expensive computation than finding resolutions just for Sylow subgroups.

The default algorithm used by the function ModPCohomologyRing() for constructing resolutions of a finite group \(G\) is ResolutionFiniteGroup() or ResolutionPrimePowerGroup() in the case when \(G\) happens to be a group of prime-power order. If the user is able to construct the first \(deg\) terms of free resolutions \(RG, RH\) for the groups \(G, H\) then the pair [RG,RH] can be entered as the third input variable of ModPCohomologyRing().

induced by the the canonical inclusion of the alternating group \(A_6\) into the symmetric group \(S_6\).

gap> G:=SymmetricGroup(6);;
gap> H:=AlternatingGroup(6);;
gap> f:=GroupHomomorphismByFunction(H,G,x->x);;
gap> RG:=ResolutionFiniteGroup(G,7);;   
gap> RH:=ResolutionFiniteSubgroup(RG,H);;
gap> F:=ModPCohomologyRing(f,2,[RG,RH]);       
[ 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 ]

8.3 Cohomology rings of finite \(2\)-groups

The following example determines a presentation for the cohomology ring \(H^\ast(Syl_2(M_{12}),\mathbb Z_2)\). The Lyndon-Hochschild-Serre spectral sequence, and Groebner basis routines from Singular (for commutative rings), are used to determine how much of a resolution to compute for the presentation.

gap> G:=SylowSubgroup(MathieuGroup(12),2);;
gap> Mod2CohomologyRingPresentation(G);
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_3^3+x_3*x_5, 
  x_1^2*x_4+x_1*x_3*x_4+x_3^2*x_4+x_3^2*x_5+x_1*x_6+x_4^2+x_4*x_5, 
  x_1^2*x_3^2+x_1*x_3*x_5+x_3^2*x_5+x_3*x_6, 
  x_1^3*x_3+x_3^2*x_4+x_3^2*x_5+x_1*x_6+x_3*x_6+x_4*x_5, 
  x_1*x_3^2*x_4+x_1*x_3*x_6+x_1*x_4*x_5+x_3*x_4^2+x_3*x_4*x_5+x_3*x_5^\
2+x_4*x_6, x_1^2*x_3*x_5+x_1*x_3^2*x_5+x_3^2*x_6+x_3*x_5^2, 
  x_3^2*x_4^2+x_3^2*x_5^2+x_1*x_5*x_6+x_3*x_4*x_6+x_4*x_5^2, 
  x_1*x_3*x_4^2+x_1*x_3*x_4*x_5+x_1*x_3*x_5^2+x_3^2*x_5^2+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_4^2*x_5+x_4*x_5^2+x_6^\
2, x_1*x_3^2*x_6+x_3^2*x_4*x_5+x_1*x_5*x_6+x_4*x_5^2, 
  x_1^2*x_3*x_6+x_1*x_5*x_6+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 ]

8.4 Steenrod operations for finite \(2\)-groups

The command CohomologicalData(G,n) prints complete information for the cohomology ring \(H^\ast(G, Z_2 )\) and steenrod operations for a \(2\)-group \(G\) provided that the integer \(n\) is at least the maximal degree of a generator or relator in a minimal set of generatoirs and relators for the ring.

The following example produces complete information on the Steenrod algebra of group number \(8\) in GAP's library of groups of order \(32\). Groebner basis routines (for commutative rings) from Singular are called in the example. (This example take over 2 hours to run. Most other groups of order 32 run significantly quicker.)

gap> CohomologicalData(SmallGroup(32,8),12);

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

8.5 Steenrod operations on the classifying space of a finite \(p\)-group

The following example constructs the first eight degrees of the mod-\(3\) cohomology ring \(H^\ast(G,\mathbb Z_3)\) for the group \(G\) number 4 in GAP's library of groups of order \(81\). It determines a minimal set of ring generators lying in degree \(\le 8\) and it evaluates the Bockstein operator on these generators. Steenrod powers for \(p\ge 3\) are not implemented as no efficient method of implementation is known.

gap> G:=SmallGroup(81,4);;
gap> A:=ModPSteenrodAlgebra(G,8);;
gap> List(ModPRingGenerators(A),x->Bockstein(A,x));
[ 0*v.1, 0*v.1, v.5, 0*v.1, (Z(3))*v.7+v.8+(Z(3))*v.9 ]

8.6 Mod-\(p\) cohomology rings of crystallographic groups

Mod \(p\) cohomology ring computations can be attempted for any group \(G\) for which we can compute sufficiently many terms of a free \(ZG\)-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.

8.6-1 Poincare series for crystallographic groups

Consider the space group \(G=SpaceGroupOnRightIT(3,226,'1')\). The following computation computes the infinite series

in which the coefficient of the monomial \(x^n\) is guaranteed to equal the dimension of the vector space \(H^n(G,\mathbb Z_2)\) in degrees \(n\le 14\). One would need to involve a theoretical argument to establish that this equality in fact holds in every degree \(n\ge 0\).

gap> G:=SpaceGroupIT(3,226);
SpaceGroupOnRightIT(3,226,'1')
gap> R:=ResolutionSpaceGroup(G,15);
Resolution of length 15 in characteristic 0 for <matrix group with 
8 generators> . 
No contracting homotopy available. 

gap> D:=List([0..14],n->Cohomology(HomToIntegersModP(R,2),n));
[ 1, 2, 5, 9, 11, 15, 20, 23, 28, 34, 38, 44, 51, 56, 63 ]

gap> PoincareSeries(D,14);
(-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)

Consider the space group \(SpaceGroupOnRightIT(3,103,'1')\). The following computation uses a different construction of a free resolution to compute the infinite series

in which the coefficient of the monomial \(x^n\) is guaranteed to equal the dimension of the vector space \(H^n(G,\mathbb Z_2)\) in degrees \(n\le 99\). The final commands show that \(G\) acts on a (cubical) cellular decomposition of \(\mathbb R^3\) with cell ctabilizers being either trivial or cyclic of order \(2\) or \(4\). From this extra calculation it follows that the cohomology is periodic in degrees greater than \(3\) and that the Poincare series is correct in every degree \(n \ge 0\).

gap> G:=SpaceGroupIT(3,103);
SpaceGroupOnRightIT(3,103,'1')
gap> R:=ResolutionCubicalCrystGroup(G,100);
Resolution of length 100 in characteristic 0 for <matrix group with 6 generators> . 

gap> D:=List([0..99],n->Cohomology(HomToIntegersModP(R,2),n));;
gap> PoincareSeries(D,99);
(x_1^3+2*x_1^2+2*x_1+1)/(-x_1+1)


#Torsion subgroups are cyclic
gap> B:=CrystGFullBasis(G);;
gap> C:=CrystGcomplex(GeneratorsOfGroup(G),B,1);;
gap> for n in [0..3] do
> for k in [1..C!.dimension(n)] do
> Print(StructureDescription(C!.stabilizer(n,k)),"  ");
> od;od;
C4  C2  C4  1  1  C4  C2  C4  1  1  1  1

8.6-2 Mod \(2\) cohomology rings of \(3\)-dimensional crystallographic groups

Computations in the integral cohomology of a crystallographic group are illustrated in Section 1.19. The commands underlying that illustration could be further developed and adapted to mod \(p\) cohomology. Indeed, the authors of the paper [LY24a] have developed commands for accessing the mod \(2\) cohomology of \(3\)-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 [LY24b]. In particular, their code contains the command

that inputs an integer in the range \(1\le ITC\le 230\) corresponding to the numbering of a \(3\)-dimensional space group \(G\) in the International Table for Crystallography. This command returns

The command SpaceGroupCohomologyRingGapInterface(ITC) is fast for most groups (a few seconds to a few minutes) but can be very slow for certain space groups (e.g. ITC \(= 228\) and ITC \(= 142\)). The following illustration assumes that two relevant files have been downloaded from [LY24b] and illustrates the command for ITC \( =30\) and ITC \(=216\).

gap> Read("SpaceGroupCohomologyData.gi");        #These two files must be 
gap> Read("SpaceGroupCohomologyFunctions.gi");   #downloaded from
gap>       #https://github.com/liuchx1993/Space-Group-Cohomology-and-LSM/
 
gap> IsPeriodicSpaceGroup(SpaceGroupIT(3,30));
true

gap> SpaceGroupCohomologyRingGapInterface(30);
===========================================
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


gap> IsPeriodicSpaceGroup(SpaceGroupIT(3,216));
false

gap> SpaceGroupCohomologyRingGapInterface(216);
===========================================
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

In the example the naming convention for ring generators follows the paper [LY24a].

8 Cohomology rings and Steenrod operations for groups

8.1 Mod-\(p\) cohomology rings of finite groups