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8 Cohomology operations
8.1 Steenrod operations on the classifying space of a finite \(2\)-group
8.2 Steenrod operations on the classifying space of a finite \(p\)-group

### 8 Cohomology operations

#### 8.1 Steenrod operations on the classifying space of a finite \(2\)-group

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, are used to determine how much of a resolution to compute for the presentation.

    gap> G:=SylowSubgroup(MathieuGroup(12),2);;
    gap> Mod2CohomologyRingPresentation(G);
    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 ]

The command `CohomologicalData(G,n)` prints complete information for the cohomology ring \(H^\ast(G, Z_2 )\) of a \(2\)-group \(G\) provided that the integer \(n\) is at least the maximal degree of a relator in a minimal set of relators for the ring. Groebner basis routines from Singular are called involved in the example.

The following example produces complete information on the Steenrod algebra of group number \(8\) in GAP's library of groups of order \(32\).

    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

#### 8.2 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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