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

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8 Cohomology rings and Steenrod operations for finite groups
8.1 Mod-\(p\) cohomology rings of finite groups
8.1-1 Ring presentations (for the commutative \(p=2\) case)
8.2 Functorial ring homomorphisms in Mod-\(p\) cohomology
8.2-1 Testing homomorphism properties
8.2-2 Testing functorial properties
8.2-3 Computing with larger groups
8.3 Cohomology rings of finite \(2\)-groups
8.4 Steenrod operations for finite \(2\)-groups
8.5 Steenrod operations on the classifying space of a finite \(p\)-group

### 8 Cohomology rings and Steenrod operations for finite groups

#### 8.1 Mod-\(p\) cohomology rings of finite 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 ]

The additional command

    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

The following example constructs the ring homomorphism

\(F\colon H^{\le deg}(G,\mathbb Z_p) \rightarrow H^{\le deg}(H,\mathbb Z_p)\)

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

The following commands are consistent with \(F\) being a ring homomorphism.

    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()`.

For instance, the following example constructs the ring homomorphism

\(F\colon H^{\le 7}(A_6,\mathbb Z_2) \rightarrow H^{\le 7}(S_6,\mathbb Z_2)\)

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 ]

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