The Cohomology of the Sylow 2 -subgroup of J_{2}

John Maginnis Mathematics Department Kansas State University Manhattan, KS 66506

Abstract. The Hall-Janko-Wales group J_{2} is one of the twenty-six sporadic finite simple groups. The cohomology of its Sylow 2 -subgroup S_{J} is computed, an important step in calculating the mod 2 cohomology of J_{2} . The spectral sequence corresponding to the central extension for S_{J} is described and shown to collapse at the eighth page. The group S_{J} contains two subgroups 2_{−}^{1+4} (the central product of a dihedral and a quaternionic group) and 2^{2+4} (the Sylow 2 -subgroup of the matrix group P SL_{3}(𝔽_{4}) ) which detect the cohomology of S_{J} . The cohomology relations for the subgroup 2^{2+4} are computed.

Notation: A presentation for S_{J} and a description of the subgroup structure appear in Gorenstein and Harada [3], but a different notation will be used here. The following is an alternate presentation for S_{J} .

S_{J} =⟨z, t, a, b, w, x, y | w^{2} = x^{2} = y^{2} = t^{2} = z^{2} = 1, a^{2} = b^{2} = z,

[w, a] = [y, b] = [a, b] = [t, x] = [a, x] = [b, x] = z, [w, b] = t, [a, y] = zt, [w, x] = a, [x, y] = b; all other commutators of the generators are trivial⟩

The group S_{J} has order 128 and a center Z(S_{J}) of order two, generated by z. The quotient S_{J}/Z(S_{J}) is isomorphic to the Sylow 2 -subgroup of the matrix group GL_{4}(𝔽_{2}) . A copy of 2_{−}^{1+4} is generated by ⟨z, t, a, b, x⟩ and a copy of 2^{2+4} is generated by ⟨z, t, a, b, w, y⟩.

When appropriate, the same symbols will be used to represent cohomology classes. For example, z ∈ H^{1}(ℤ/2ℤ; 𝔽_{2}) is a ring generator for the cohomology of the center of S_{J} , and w ∈ H^{1}(S_{J}; 𝔽_{2}) represents the homomorphism S_{J} → ℤ/2ℤ with kernel generated by ⟨z, t, a, b, x, y⟩. It should be clear from context whether a symbol denotes a group element or a cohomology class.

The two detecting subgroups: The computation of the cohomology of the extra special 2 - group 2_{−}^{1+4} is due to Quillen [6].

Lemma 1 The mod 2 cohomology of 2_{−}^{1+4} is generated by four 1 -dimensional classes t, a, b, x, and one 8 -dimensional class Z_{8} . The relations are spanned by a^{2} = ab + b^{2} + tx + ax + bx, b^{3} = b^{2}x + tbx + t^{2}x, and t^{4}x = t^{3}x^{2} .

The second and third relations are usually written as the Steenrod squares Sq^{1} and Sq^{2}Sq^{1} applied to the first relation, but the given relations are equivalent.

The following is a result of Adem and Milgram [1].

Lemma 2 The mod 2 cohomology of 2^{2+4} is generated by four 1 -dimensional classes w, y, a, b, two 3 -dimensional classes L_{1} , L_{2} , two 4 -dimensional classes T_{4} , Z_{4} , and one 6 -dimensional class R_{2} .

For later use, a description of two spectral sequences converging to the cohomology of 2^{2+4} will be given. Adem and Milgram use the central group extension

2^{2} → 2^{2+4} → 2^{4}.  (1)

The d^{2} -differential of the corresponding spectral sequence is determined by d^{2}(t) = wb + ay and d^{2}(z) = a^{2} + ab + b^{2} + aw + by + ay. Note that the row E_{3}^{∗,1} = 0 . The d^{3} -differential is then given by d^{3}(t^{2}) = b^{2}y + by^{2} and d^{3}(z^{2}) = b^{3} + b^{2}y + aby + ay^{2} . The permanent classes L_{1} = z^{2}w + t^{2}(a + w + y) and L_{2} = z^{2}y + t^{2}(b + w) arise on this page. Note the differentials d^{3}(t^{2}w) = aby^{2} + ay^{3} and d^{3}(z^{2}t^{2}) = (b^{2} + by)L_{2} . We also have classes R_{1} = z^{2}t^{2}(w^{2} + wy + y^{2}) and R_{2} = z^{2}t^{2}(aw+by+ay) , although R_{1} detects the product L_{1}L_{2} and is not a new generator. The spectral sequence then collapses, with permanent classes T_{4} = t^{4} and Z_{4} = z^{4} .

We obtain an additive description of the cohomology:

\begin{aligned}H^{∗}(2^{2+4}; 𝔽_{2}) & = 𝔽_{2}[T_{4}, Z_{4}] ⊗ \{ab + b^{2} + ab^{2} + aby + (1 + L_{1} + L_{2} + L_{1}L_{2}) \\ & \{(1 + a)𝔽_{2}[w, y] + b𝔽_{2}[y]\} + abL_{2} + R_{2}\} . \end{aligned}  (2)

We now give another description of 2^{2+4} as a group extension. The quotient 2^{2+4}/⟨z⟩ is isomorphic to 2_{+}^{1+4} , the central product of two copies of the dihedral group.

Lemma 3 [6] The mod 2 cohomology of 2_{+}^{1+4} is generated by four 1 -dimensional classes w, y, a, b, and one 4 -dimensional class T_{4} . The relations are spanned by wb = ay and w^{2}b + wb^{2} = a^{2}y + ay^{2} .

The spectral sequence corresponding to the group extension

2 → 2^{2+4} → 2_{+}^{1+4}  (3)

has d^{2} -differential determined by d^{2}(z) = a^{2} + ab + b^{2} + aw + by + ay. Note that d^{2}(zw) = a^{2}w + aw^{2} , d^{2}(zy) = b^{2}y + by^{2} , d^{2}(zwy) = aby^{2} + ay^{3} , and d^{2}(zay) = ab^{2}y + aby^{2} . The kernel of d^{2} is generated by N_{1} = zwy(w + y) and N_{2} = zay(w + y) . The row E_{3}^{∗,1} may be described either as

(N_{1} + N_{2})𝔽_{2}[w, y, T_{4}] or as 𝔽_{2}[T_{4}] ⊗ {N_{1}(1 + a)𝔽_{2}[w, y] + N_{2}𝔽_{2}[y]}.

The d^{3} -differential is determined by d^{3}(z^{2}) = a^{2}b + ab^{2} + a^{2}y + ay^{2} . The kernel of d^{3} is generated by L_{1} = z^{2}w and L_{2} = z^{2}y, and the row E_{4}^{∗,2} = 𝔽_{2}[T_{4}] ⊗ {L_{1}(1 + a)𝔽_{2}[w, y] + L_{2}(ab + (1 + a + b)𝔽_{2}[y]}. We must also consider d^{3}(z^{2}N_{1}) and d^{3}(z^{2}N_{2}) , but both are zero. By comparing with the results of Adem and Milgram, or by restricting to various subgroups of 2^{2+4} , we find that d^{4}(z^{2}N_{1}) = d^{4}(z^{2}N_{2}) = 0 . Thus R_{1} = z^{3}wy(w + y) and R_{2} = z^{3}ay(w + y) are permanent classes.

The spectral sequence now collapses, with Z_{4} = z^{4} . It can be shown that N_{1} = yL_{1} + wL_{2} , N_{2} = bL_{1} + aL_{2} , and R_{1} detects the product L_{1}L_{2} .

The cohomology of S_{J} : The central quotient S_{J}/Z(S_{J}) is isomorphic to the Sylow 2 - subgroup of GL_{4}(𝔽_{2}) . The cohomology of Syl_{2}(GL_{4}(𝔽_{2})) was originally computed by Tezuka and Yagita [7], although a more complete description appears in the author’s thesis [4].

Lemma 4 [4] The mod 2 cohomology of Syl_{2}(GL_{4}(𝔽_{2})) is generated by three 1 -dimensional classes w, x, y, three 2 -dimensional classes α, β , d, one 3 -dimensional class D, and one 4 -dimensional class T_{4} . The relations are spanned by wx = xy = 0 , wd = yd = wβ = yα, wD = yD = w^{2}β , d^{2} = xD + x^{2}d + αβ , and

D^{2} = x^{2}T_{4} + xD(d + α + β) + x^{2}d(α + β) + αβ(α + β) + w^{2}β^{2}.

The Steenrod squaring operations are Sq^{1}(α) = (w + x)α, Sq^{1}(β) = (x + y)β , Sq^{1}(d) = D, Sq^{2}(D) = xT_{4} + D(d + α + β + x^{2}) + xd(α + β) + w^{3}β , Sq^{1}(T_{4}) = xT_{4} , and Sq^{2}(T_{4}) = (α + β + d + w^{2} + y^{2})T_{4} .

Lemma 5 [4] For 2^{4} ⊂ Syl_{2}(GL_{4}(𝔽_{2})) the restriction map satisfies

Res(α) = a^{2} + ax
Res(β) = b^{2} + bx
Res(d) = tx + ab
Res(T_{4}) = t^{4} + t^{3}x + t^{2}(ab + a^{2} + ax + b^{2} + bx) + t(a^{2}b + ab^{2} + abx).

Lemma 6 [4] For 2_{+}^{1+4} ⊂ Syl_{2}(GL_{4}(𝔽_{2})) the restriction map satisfies

Res(α) = a^{2} + aw
Res(β) = b^{2} + by
Res(d) = ab + ay.

Theorem 7 The mod 2 cohomology of S_{J} is generated by three 1 -dimensional classes w, x, y, two 2 -dimensional classes α, β , two 3 -dimensional classes L_{1} , L_{2} , one 4 -dimensional class T_{4} , two 5 -dimensional classes M_{1} , M_{2} , one 8 -dimensional class Z_{8} , and one 10 -dimensional class V .

Proof: We obtain the generators and filtered relations by computing the spectal sequence for the central extension

2 → S_{J} → Syl_{2}(GL_{4}(𝔽_{2})).  (4)

The various pages of the sequence are described additively.

E_{2}^{∗,0} = H^{∗}(Syl_{2}(GL_{4}(𝔽_{2})); 𝔽_{2})
= (1 + d + D + dD)𝔽_{2}[x, α, β, T_{4}] + wβ𝔽_{2}[w, β, T_{4}]
+ w𝔽_{2}[w, α, T_{4}] + y𝔽_{2}[y, β, T_{4}] + wy𝔽_{2}[w, y, T_{4}].

The d^{2} -differential is determined by d^{2}(z) = d + α + β . Note that d^{2}(zw) = wα, d^{2}(zy) = yβ , d^{2}(zwy) = w^{2}β , d^{2}(zwβ) = wβ^{2} , d^{2}(zd) = xD +x^{2}(α +β)+α^{2} +αβ +β^{2} , and d^{2}(zdD) = x^{3}T_{4} + (α^{2} + αβ + β^{2})D + x(α + β)^{3} . The kernel of the d^{2} -differential is generated by N_{1} = zwy(w + y) .

E_{3}^{∗,1} = N_{1}𝔽_{2}[w, y, T_{4}].

We find that E_{3}^{∗,1} = E_{∞}^{∗,1} since this row is detected on an elementary abelian subgroup 2^{4} ⊂ S_{J} , with 2^{4} generated by the group elements ⟨z, t, w, y⟩.

\begin{aligned}E_{3}^{∗,0} & = 𝔽_{2}[x, α, β] + (1 + x + x^{2})T_{4}𝔽_{2}[α, β, T_{4}] \\ & + D𝔽_{2}[α, β, T_{4}] + wβ𝔽_{2}[T_{4}] + w𝔽_{2}[w, y, T_{4}] + y𝔽_{2}[y, T_{4}]. \end{aligned}

The d^{3} -differential is determined by d^{3}(z^{2}) = D+x(α+β) . Note that d^{3}(z^{2}x) = α^{2}+αβ+β^{2} and d^{3}(z^{2}D) = x^{2}T_{4} + αβ(α + β) . We find the permanent classes L_{1} = z^{2}w and L_{2} = z^{2}y. By restriction to the subgroup 2^{2+4} , we find that R_{1} = z^{2}N_{1} = z^{3}wy(w+y) represents a permanent class, and also that N_{1} = yL_{1} + wL_{2} .

E_{5}^{∗,3} = R_{1}𝔽_{2}[w, y, T_{4}].
E_{5}^{∗,2} = L_{1}β𝔽_{2}[T_{4}] + L_{1}𝔽_{2}[w, y, T_{4}] + L_{2}𝔽_{2}[y, T_{4}].
E_{5}^{∗,0} = (1 + α)𝔽_{2}[x, β] + (1 + x)(1 + α)T_{4}𝔽_{2}[β, T_{4}]
+ wβ𝔽_{2}[T_{4}] + w𝔽_{2}[w, y, T_{4}] + y𝔽_{2}[y, T_{4}].
Note that E_{5}^{∗,3} = E_{∞}^{∗,3} , detected on 2^{4} .

The d^{5} -differential is determined by d^{5}(z^{4}) = xT_{4} . Note that d^{5}(z^{4}x) = αβ(α + β) , which yields α^{3} = 0 and β^{3} = 0 . We have permanent classes M_{1} = z^{4}w and M_{2} = z^{4}y. By restricting to 2^{2+4} , we see that R_{1} detects L_{1}L_{2} + yM_{1} . We must also consider the differentials on the classes z^{4}L_{1} , z^{4}L_{2} , Q = z^{4}N_{1} , and V = z^{4}R_{1} . Since N_{1} = yL_{1} + wL_{2} , we have Q = L_{1}M_{2} + L_{2}M_{1} is a permanent class.

I claim that there are d^{7} -differentials d^{7}(z^{4}L_{1}) = α^{2}T_{4} and d^{7}(z^{4}L_{2}) = β^{2}T_{4} , and that L_{1}M_{1} , L_{1}M_{2} , and L_{2}M_{2} survive as permanent classes. This can be seen in three different ways. First, we have d^{5}(z^{4}L_{1}) = xL_{1}T_{4} . But xL_{1} is hit by a d^{1} -differential. Since E_{3}^{∗,1} = E_{∞}^{∗,1} as detected on 2^{4} , we must have d^{6} = 0 . We will compute the relations in the mod 2 cohomology of S_{J} , finding xL_{1} = α^{2} and xL_{2} = β^{2} . So of course xL_{1}T_{4} = α^{2}T_{4} . The fact that representative cocyles for these two classes are cohomologous is detected on the d^{1} -differential. This will imply that d^{7}(z^{4}L_{1}) = α^{2}T_{4} . Note that the relation α^{2}T_{4} = 0 detected by this d^{7} -differential follows from xT_{4} = 0 , given by the transgression d^{5}(z^{4}) , along with xL_{1} = α^{2} .

Alternatively, the d^{7} -differential is detected on the subgroup 2_{−}^{1+4} as a d^{4} -differential. This requires computing the restrictions of L_{1} and L_{2} to 2_{−}^{1+4} .

Lastly, it is impossible for d^{7} = 0 . We will compute the long exact Gysin sequence for 2^{2+4} ⊂ S_{J} , and if z^{4}L_{1} were a permanent class, the restriction to 2^{2+4} would involve the class Z_{4}L_{1} . We will see that no such restriction exists by considering the transfer map.

In addition, the Gysin sequence will show the existence of a class that restricts to 2^{2+4} to a class involving Z_{4}R_{1} , which should be detected on our spectral sequence (4) by V = z^{4}R_{1} . Alternatively, the classes that a differential on V could possibly hit can all be shown to be non-zero by detection on 2^{2+4} or 2_{−}^{1+4} .

The spectral sequence (4) now collapses, with the permanent class Z_{8} = z^{8} . Note that Z_{8} can be described as an Evens norm [2] from the subgroup 2^{4} ⊂ S_{J} , of index 8 . Alternatively, Z_{8} is a Stiefel-Whitney class of a representation induced from one for 2^{4} .

The mod 2 cohomology of S_{J} can be described as

\begin{aligned}H^{∗}(S_{J}; 𝔽_{2}) & = 𝔽_{2}[Z_{8}] ⊗ \{(1 + α)(1 + β + β^{2})𝔽_{2}[x] \\ & + 𝔽_{2}[T_{4}] ⊗ \{(1 + α + β)T_{4} + β(w + L_{1} + M_{1}) \\ & + (w + L_{1} + L_{2} + L_{1}L_{2} + M_{1} + L_{1}M_{1} + L_{2}M_{1} + V)𝔽_{2}[w, y] \\ & + (y + M_{2} + L_{1}M_{2} + L_{2}M_{2})𝔽_{2}[y]\}\}. \end{aligned}  (5)

Gysin sequences: For any index two subgroup H ⊂ G, there is a long exact sequence involving restriction and transfer maps. Let g ∈ H^{1}(G; 𝔽_{2}) denote the cohomology class represented by the homomorphism G → G/H . The Gysin sequence is

⋯ H^{q}(G; 𝔽_{2}) →^{Res} H^{q}(H; 𝔽_{2}) →^{Tr} H^{q}(G; 𝔽_{2}) →^{−∪ g} H^{q+1}(G; 𝔽_{2}) ⋯

The transfer map T r commutes with the Steenrod squaring operations, and satisfies T r(A ∪ Res(B)) = T r(A) ∪ B .

Theorem 8 For 2^{2+4} ⊂ S_{J} the restriction and transfer maps satisfy

Res(w) = w
Res(x) = 0
Res(y) = y
Res(L_{1}) = L_{1}
Res(L_{2}) = L_{2}
Res(T_{4}) = T_{4}

Res(α) = a^{2} + aw
Res(β) = b^{2} + by
Res(M_{1}) = wZ_{4} + aT_{4}
Res(M_{2}) = yZ_{4} + bT_{4}
Res(Z_{8}) = Z_{4}^{2} + Z_{4}T_{4}
Res(V) = Z_{4}R_{1} + T_{4}R_{2}
\begin{aligned}T r(a) & = w \\ T r(b) & = y \end{aligned}  \begin{matrix}T r(Z_{4}) = T_{4} \\ T r(R_{2}) = R_{1} \end{matrix}

Proof: Consider the Gysin sequence for this index two subgroup. An application of the Double Coset Formula shows that the composition Res ∘ T r = 1 + c_{x} , where assuming trivial coefficients c_{x} is induced by conjugation by the group element x ∈ S_{J} . Dualizing this conjugation to obtain the action on cohomology yields the transfers T r(a) = w and T r(b) = y. Recall that T r(Sq^{n}(A)) = Sq^{n}(T r(A)) and T r(A ∪ Res(B)) = T r(A) ∪ B . Thus T r(a^{2} + aw) = 0 and T r(b^{2} + by) = 0 , and we denote by α and β the classes with the appropriate restrictions.

The spectral sequence (4) used to compute the cohomology of S_{J} has a d^{5} -differential with image xT_{4} . Thus in the Gysin sequence, T_{4} is in the kernel of the cup product with the class x, and in the image of the transfer. Let Z_{4} be a class satisfying T r(Z_{4}) = T_{4} . We then obtain T r(wZ_{4} + aT_{4}) = 0 , T r(yZ_{4} + bT_{4}) = 0 and T r(Z_{4}^{2} + Z_{4}T_{4}) = 0 . Denote by M_{1} , M_{2} and Z_{8} classes having the respective restrictions.

The class R_{2} ∈ H^{∗}(2^{2+4}; 𝔽_{2}) is detected on the two spectral sequences used to describe the cohomology of 2^{2+4} by the filtered names R_{2} = z^{2}t^{2}(aw + by + ay) or R_{2} = z^{3}ay(w + y) . Also recall the filtered names R_{1} = z^{2}t^{2}(w^{2} + wy + y^{2}) or R_{1} = z^{3}wy(w + y) . Conjugation by the group element x induces an action on each spectral sequence, satisfying c_{x}(R_{2}) = R_{1} + R_{2} . Given a choice of the class R_{2} (only determined up to linear combination of the other generators), define R_{1} ∈ H^{∗}(S_{J}; 𝔽_{2}) by R_{1} = T r(R_{2}) . Note R_{1} detects L_{1}L_{2} +yM_{1} , but may involve other classes as well. Also denote the restriction Res(R_{1}) by R_{1} . Then T r(Z_{4}R_{1} + T_{4}R_{2}) = 0 , and denote by V a class with the appropriate restriction. Note that V will indeed be detected on the spectral sequence (4) as z^{4}R_{1} .

Corollary 9 The kernel of the restriction map for 2^{2+4} ⊂ S_{J} is

(x + αβ + β^{2} + αβ^{2})𝔽_{2}[x, Z_{8}].

Proof: The Gysin sequence states that the kernel consists of precisely those classes which are multiples of x. It will eventually be shown that xL_{1} = α^{2} and xL_{2} = β^{2} , so that αβ = α^{2} +β^{2} = x(L_{1} +L_{2}) . For now, note that the transgressions in the spectral sequence (1) for 2^{2+4} imply the relations

\begin{aligned}wb & = ay \\ a^{2} & = ab + b^{2} + aw + by + ay \\ b^{2}y & = by^{2} \\ b^{3} & = by^{2} + aby + ay^{2} \\ aby^{2} & = ay^{3}. \end{aligned}  (6)

Therefore

Res(β^{2}) = (b^{2} + by)^{2} = b^{4} + b^{2}y^{2}
= b(by^{2} + aby + ay^{2}) + b^{2}y^{2}
= ab^{2}y + aby^{2}
= 0,

Res(αβ) = (a^{2} + aw)(b^{2} + by)
= (ab + b^{2} + by + ay)(b^{2} + by)
= b^{3}(a + b) + b^{2}y^{2} + aby^{2}
= (by^{2} + aby + ay^{2})(a + b) + b^{2}y^{2} + aby^{2}
= a^{2}by + a^{2}y^{2} + ab^{2}y + aby^{2}
= (ab + b^{2} + aw + by + ay)by + a^{2}y^{2} + ab^{2}y + aby^{2}
= abwy + b^{3}y + b^{2}y^{2} + a^{2}y^{2}
= a^{2}y^{2} + b(b^{2}y + by^{2}) + a^{2}y^{2}
= 0.

To see that the kernel is no larger, compare the cohomology descriptions (2) and (5). Note that the restriction of α is detected by the class ab. Also, Res(wβ) = wb^{2} + wby = aby + ay^{2} , and the restriction of βM_{1} is detected by abyZ_{4} . Considering the filtered descriptions in the spectral sequences (3) and (4), L_{1} = z^{2}w and L_{2} ≡ z^{2}y, we find that the restriction of βL_{1} is detected by (ab + ay)L_{2} .

Theorem 10 For 2_{−}^{1+4} ⊂ S_{J} , the restriction map satisfies

Res(w) = 0
Res(x) = x
Res(y) = 0
Res(β) = b^{2} + bx
Res(T_{4}) = t^{4} + t^{3}x
Res(Z_{8}) = Z_{8}

Res(α) = tx + ab + b^{2} + bx
Res(L_{1}) = t^{2}a + tab + tb^{2} + tbx
Res(L_{2}) = t^{2}b + t^{2}x + tb^{2} + tbx
Res(M_{1}) = a(t^{4} + t^{3}x)
Res(M_{2}) = b(t^{4} + t^{3}x)

Remarks: We must leave the computation of the restriction of the class V until later, but I claim Res(V) = ab^{2}t^{3}(t^{4} + t^{3}x) . The proof of Theorem 10 will require several lemmas.

Gorenstein and Harada [3, Lemma 5.1] denote a certain index two subgroup of S_{J} by T_{2} , with T_{2} generated by ⟨z, t, a, b, x, y⟩. A conjugate of this subgroup is T′_{2} = ⟨z, t, a, b, w, x⟩. Note that T_{2} ∩ T′_{2} = 2_{−}^{1+4} .

The center of T_{2} is 2 = ⟨z⟩, and the central quotient T_{2}/⟨z⟩ is isomorphic to the wreath product 2^{2} ∫ 2 = 2^{4} ×_{T} 2 .

Lemma 11 The mod 2 cohomology of 2^{2} ∫ 2 is generated by three 1 -dimensional classes a, x, y, and three 2 -dimensional classes d, β , τ . The relations are spanned by ay = xy = yd = 0 and d^{2} = axd + a^{2}β + x^{2}τ . The Steenrod squaring operations are determined by Sq^{1}(β) = (x + y)β , Sq^{1}(τ) = (a + y)τ , and Sq^{1}(d) = (a + x)d + aβ + xτ .

Proof: See [4]. Also see Nakaoka [5], concerning the cohomology of a wreath product.

Lemma 12 The mod 2 cohomology of T_{2} is generated by three 1 -dimensional classes a, x, y, two 2 -dimensional classes β , τ , one 3 -dimensional class L_{2} , one 5 -dimensional class M_{2} , and one 8 -dimensional class Z_{8} .

Proof: The spectral sequence for the group extension

2 → T_{2} → 2^{2} ∫ 2  (7)

has d^{2} -differential determined by d^{2}(z) = d + β + a^{2} + ax. Note that d^{2}(zy) = yβ and d^{2}(zd + zβ + za^{2}) = x^{2}τ + β^{2} + a^{2}β + axβ + a^{4} + a^{3}x. The d^{2} -differential has no kernel. The d^{3} -differential is given by d^{3}(z^{2}) = xτ + a^{3} + a^{2}x, and there is a permanent class L_{2} = z^{2}y. Then d^{5}(z^{4}) = xτ^{2} + xa^{2}τ + x^{2}aτ , and the class M_{2} = z^{4}y survives. There is a d^{7} -differential, as in the S_{J} situation, d^{7}(z^{4}L_{2}) = β^{2}(τ^{2} + a^{2}τ + axτ) . Note that L_{2}M_{2} survives, and the spectral sequence collapses with Z_{8} = z^{8} .

Lemma 13 For T_{2} ⊂ S_{J} , the restriction and transfer maps satisfy

\begin{aligned}T r(a) & = x \\ T r(τ) & = α \end{aligned}  ^{T r(aτ) = xα + wβ}

Res(w) = 0
Res(x) = x
Res(y) = y
Res(β) = β
Res(L_{2}) = L_{2}
Res(M_{2}) = M_{2}

Res(α) = a^{2} + ax
Res(L_{1}) = (a + x + y)τ
Res(T_{4}) = τ^{2} + (a^{2} + ax)τ
Res(M_{1}) = aτ^{2} + (a^{3} + a^{2}x)τ
Res(Z_{8}) = Z_{8}

Proof: The spectral sequence (4) for S_{J} yields the relations wx = 0 and wα = 0 . The Gysin sequence for T_{2} ⊂ S_{J} then implies that x and α are in the image of the transfer map. We must have T r(a) = x; then T r(a^{2}) = T r(ax) = x^{2} , and so Res(α) = a^{2} + ax. T r(τ) is either α or α + x^{2} . The relation xτ = a^{3} + a^{2}x, from the spectral sequence (7) for T_{2} , yields T r(xτ) = T r(a(a^{2} + ax)) = αT r(a) = xα. Therefore T r(τ) = α. Similarly, to compute T r(aτ) we consider T r(axτ) = T r(a^{4} + a^{3}x) = T r((a^{2} + ax)^{2} + ax(a^{2} + ax)) = x^{2}α. Thus T r(aτ) = aα + A with xA = wA = 0 . Either A = 0 or A = wβ .

The filtered description L_{1} = z^{2}w + t^{2}(a + w + y) from the spectral sequence (1) implies that Res(L_{1}) involves (a + y)τ . Note that T r((a + y)τ) = T r(aτ) + yα, and yα = wβ . Since any other class contained in Res(L_{1}) would yield transfers which are multiples of x, we must have A = wβ . Alternatively, one could use the double coset formula for Res_{2^{2+4}}^{S_{J}}T r_{T_{2}}^{S_{J}} to show that A = wβ .

We now have T r((a + x + y)τ) = 0 and Res(L_{1}) = (a + x + y)τ . The restriction of T_{4} follows from T r(τ^{2}) = T r((a^{2} + ax)τ) = α^{2} . Note that this calculation does not involve any relations derived from the differentials in the spectral sequence (7) ; in other words this description agrees with that inflated from Syl_{2}(GL_{4}(𝔽_{2})) . The restriction of M_{1} follows from T r(a ∪ Res(T_{4})) = xT_{4} = 0 . The description of M_{1} agrees with that given in Theorem 8.

Proof of Theorem 10: We have Res(α) = a^{2}+ax, but this is equal to tx+ab+b^{2}+bx in 2_{−}^{1+4} . A similar sequence of lemmas involving the conjugate subgroup T′_{2} implies that Res(β) = b^{2}+bx. The relation ay = 0 in H^{2}(T_{2}; 𝔽_{2}) implies that T r_{2_{−}^{1+4}}^{T_{2}}(t) = a, and so Res_{2_{−}^{1+4}}^{T_{2}}(τ) = t^{2} + ta. Therefore Res(L_{1}) = (t^{2} + ta)(a + x) , which equals t^{2}a + tab + tb^{2} + tbx. The analogous result is Res(L_{2}) = (t^{2} + tb)(b + x) . We also have Res(T_{4}) = (t^{2} + ta)^{2} + (a^{2} + ax)(t^{2} + ta) , which equals t^{4} + t^{3}x. It follows that Res(M_{1}) = a(t^{4} + t^{3}x) , and analogously Res(M_{2}) = b(t^{4} + t^{3}x) .

Note that the restriction of T_{4} agrees with that given in lemma 5, after applying the relations for H^{∗}(2_{−}^{1+4}; 𝔽_{2}) in Lemma 1.

Theorem 14 The mod 2 cohomology of S_{J} is detected on the two subgroups 2^{2+4} and 2_{−}^{1+4} ; that is we have an injection

H^{∗}(S_{J}; 𝔽_{2}) ⊂ H^{∗}(2^{2+4}; 𝔽_{2}) ⊕ H^{∗}(2_{−}^{1+4}; 𝔽_{2}).

Proof: Corollary 9 identifies the kernel of the restriction to 2^{2+4} , and this can be shown to be detected on 2_{−}^{1+4} by Theorem 10. In particular, the restriction map for 2_{−}^{1+4} ⊂ S_{J} satisfies

Res(αβ) = t^{2}(ax + bx + x^{2}) + tabx
Res(β^{2}) = t^{2}(bx + x^{2}) + t(b^{2}x + bx^{2})
Res(αβ^{2}) = t^{3}(ax^{2} + bx^{2} + x^{3}) + t^{2}(ab^{2}x + b^{2}x^{2} + bx^{3}).

Remark: Note that Res(β^{2}) = x(t^{2}b + t^{2}x + tb^{2} + tbx) = Res(xL_{2}) . Similarly, Res(α^{2}) = Res(xL_{1}) . The spectral sequence for the central extension for 2_{−}^{1+4} has d^{5} -differential determined by d^{5}(z^{4}) = x(t^{4} + t^{3}x) . Thus d^{5}(z^{4}Res(L_{1})) = xRes(L_{1})(t^{4} + t^{3}x) = Res(α^{2})(t^{4} + t^{3}x) = Res(α^{2}T_{4}) . Similarly, d^{5}(z^{4}Res(L_{2})) = Res(β^{2}T_{4}) . This verifies the d^{7} -differentials in the spectral sequence (4) , d^{7}(z^{4}L_{1}) = α^{2}T_{4} and d^{7}(z^{4}L_{2}) = β^{2}T_{4} .

Cohomology relations for S_{J} : The spectral sequence (4) for S_{J} yields the relations in H^{∗}(S_{J}; 𝔽_{2})

\begin{aligned}wx & = 0 \\ wα & = 0 \\ wβ & = yα \\ xT_{4} & = 0 \\ α^{2}T_{4} & = 0 \end{aligned}  \begin{aligned}xy & = 0 \\ yβ & = 0 \\ αβ & = α^{2} + β^{2} \\ α^{3} & = 0 \\ β^{2}T_{4} & = 0. \end{aligned}  (8)

We have already commented on the relations

xL_{1} = α^{2} xL_{2} = β^{2}.

The remaining relations may be found in filtered form from spectral sequence (4) . The complete relations may be determined from the detection theorem 12 once the relations in H^{∗}(2^{2+4}; 𝔽_{2}) are computed.

Subgroups of 2^{2+4} : In order to compute the relations in the mod 2 cohomology of 2^{2+4} , it is necessary to understand the cohomology of several subgroups. Let us use the notations

\begin{aligned}H_{a} & = ⟨z, t, b, w, y⟩ \\ H_{y} & = ⟨z, t, a, b, w⟩ \\ H_{ay} & = ⟨z, t, b, w⟩ \\ H_{aw} & = ⟨z, t, b, y⟩ = 2 × D_{8} \\ H_{by} & = ⟨z, t, a, w⟩ = 2 × D_{8} \\ H_{wy} & = ⟨z, t, a, b⟩ = 2 × Q_{8} \\ H_{ab} & = ⟨z, t, w, y⟩ = 2^{4}. \end{aligned}

Clearly the cohomology of the elementary abelian group H_{ab} is 𝔽_{2}[z, t, w, y] . The cohomologies of the dihedral group D_{8} and the quaternion group Q_{8} are standard results.

Lemma 15

(a) H^{∗}(H_{aw}; 𝔽_{2}) = 𝔽_{2}[t, b, y, Z_{2}]/(b^{2} + by)

(b) H^{∗}(H_{by}; 𝔽_{2}) = 𝔽_{2}[t, a, w, Z_{2}]/(a^{2} + aw)

(c) H^{∗}(H_{wy}; 𝔽_{2}) = 𝔽_{2}[t, a, b, Z_{4}]/(a^{2} + ab + b^{2}, a^{2}b + ab^{2})

Lemma 16 The mod 2 cohomology of H_{ay} is generated by two 1 -dimensional classes w, b, and three 2 -dimensional classes τ_{2} , γ_{2} , Z_{2} .

Proof: The spectral sequence corresponding to the extension

2 → H_{ay} → D_{8}

has d^{2} -differential d^{2}(z) = b^{2} , with the cohomology of the quotient H^{∗}(D_{8}; 𝔽_{2}) = 𝔽_{2}[w, b, τ_{2}]/(wb) . Thus γ_{2} = zw, and the spectral sequence collapses with Z_{2} = z^{2} .

Lemma 17 The mod 2 cohomology of H_{a} is generated by three 1 -dimensional classes w, y, b, and three 2 -dimensional classes τ_{2} , γ_{2} , Z_{2} . The relations are spanned by wb = 0 , b^{2} = by, bγ_{2} = 0 , and γ_{2}^{2} = w^{2}Z_{2} + wyγ_{2} + (by + y^{2})τ_{2} .

Proof: The quotient H_{a}/⟨z⟩ is isomorphic to 2 × D_{8} , and the cohomology of the quotient H^{∗}(2 × D_{8}; 𝔽_{2}) = 𝔽_{2}[w, y, b, τ_{2}]/(wb) . The spectral sequence corresponding to this extension

2 → H_{a} → 2 × D_{8}

has d^{2} -differential d^{2}(z) = b^{2} + by. Then γ_{2} = zw is a permanent class, and the spectral sequence collapses with Z_{2} = z^{2} . The relations wb = 0 and b^{2} = by follow immediately. We also have filtered relations bγ_{2} = 0 and γ_{2}^{2} = w^{2}Z_{2} . The cohomology of H_{a} can be seen to be detected on the two subgroups H_{aw} and H_{ab} , so the complete relations can be determined.

For H_{aw} ⊂ H_{a} the transfer map must satisfy T r(t) = b. Thus T r(t^{2} + tb) = 0 and T r(tb + ty) = b^{2} + by = 0 . We choose classes τ_{2} and γ_{2} to have the appropriate restrictions Res(τ_{2}) = t^{2} +tb and Res(γ_{2}) = tb+ty. Also note that for 2^{3} = ⟨z, t, y⟩ we have Res_{2^{3}}^{H_{a}}(τ_{2}) = t^{2} and Res_{2^{3}}^{H_{a}}(γ_{2}) = ty. The transfer map T r_{2^{3}}^{H_{aw}}(z) = b+y, and so T r_{2^{3}}^{H_{aw}}(z^{2} +zy) = b^{2} +by = 0 . Denote by Z_{2} the class with Res_{2^{3}}^{H_{aw}}(Z_{2}) = z^{2} + zy.

Now consider the transfer map for H_{ab} ⊂ H_{a} . The classes w and b + y must be in the image. By the double coset formula, ResT r = 1 + c_{b} , with c_{b} induced by conjugation by the group element b. The commutators [w, b] = t and [y, b] = z imply that (1 + c_{b})t = w and (1 + c_{b})z = y. We must have T r(t) = w and T r(z) = b + y. Then T r(t^{2} + tw) = 0 , T r(z^{2}+zy) = b^{2}+by = 0 , and T r(zw+ty) = wb = 0 . By restricting to the subgroup 2^{3} , factoring through H_{aw} , we find that Res(τ_{2}) = t^{2} + tw, Res(Z_{2}) = z^{2} + zy, and Res(γ_{2}) = zw + ty. Therefore Res(γ_{2}^{2} + w^{2}Z_{2} + wyγ_{2} + y^{2}τ_{2}) = 0 .

Lemma 18 The mod 2 cohomology of H_{y} is generated by three 1 -dimensional classes w, a, b, one 2 -dimensional class τ_{2} , one 3 -dimensional class L_{1} , and one 4 -dimensional class Z_{4} . The relations are spanned by wb = 0 , a^{2} = ab + b^{2} + aw, b^{3} = 0 , bL_{1} = abτ_{2} , and L_{1}^{2} = w^{2}Z_{4} + a^{2}τ_{2}^{2} + w^{2}τ_{2}^{2} .

Proof: There is a description of H_{y} as a group extension

2 → H_{y} → 2 × D_{8}

where the quotient has cohomology H^{∗}(2 × D_{8}; 𝔽_{2}) = 𝔽_{2}[w, a, b, τ_{2}]/(wb) . The d^{2} -differential is d^{2}(z) = a^{2} + ab + b^{2} + aw. Note that d^{2}(zw) = a^{2}w + aw^{2} . Then d^{3}(z^{2}) = a^{2}b + ab^{2} = b^{3} , and there is a permanent class L_{1} = z^{2}w. The spectral sequence collapses with Z_{4} = z^{4} .

The relations wb = 0 , a^{2} = ab + b^{2} + aw, and b^{3} = 0 follow immediately. We also have filtered relations bL_{1} = 0 and L_{1}^{2} = w^{2}Z_{4} . The cohomology of H_{y} is detected on the two subgroups H_{wy} and H_{by} .

For H_{wy} ⊂ H_{y} , the transfer map must satisfy T r(t) = b. Then T r(t^{2} + tb) = 0 and T r(tb^{2}) = b^{3} = 0 . Denote by τ_{2} the class having restriction Res(τ_{2}) = t^{2} + tb. Denote by L_{1} the class having restriction Res(L_{1}) = a(t^{2} + tb) + tb^{2} . This agrees with the more complete filtered description for L_{1} that would be obtained from the central extension 2^{2} → H_{y} → 2^{3} . Note that Res(bL_{1} + abτ_{2}) = b(t^{2}a + tab + tb^{2}) + ab(t^{2} + tb) = tb^{3} = 0 , and Res(L_{1}^{2} + a^{2}τ^{2}) = (t^{4}a^{2} + t^{2}a^{2}b^{2} + tb^{4}) + a^{2}(t^{4} + t^{2}b^{2}) = 0 .

For H_{by} ⊂ H_{y} , the classes w and b^{2} must be in the image of the transfer map. Thus T r(t) = w, and so T r(t^{2} + tw) = 0 and Res(τ_{2}) = t^{2} + tw. Denote by Z_{2} a class satisfying T r(Z_{2}) = b^{2} . Then T r(Z_{2}^{2}) = b^{4} = 0 and T r(Z_{2}w) = wb^{2} = 0 . Denote by Z_{4} and L_{1} the classes with restrictions Res(Z_{4}) = Z_{2}^{2} and Res(L_{1}) = Z_{2}w+a(t^{2} +tw)+w(t^{2} +tw) . This agrees with the choices made when restricting to H_{wy} . Note that ⟨z, t, a⟩ = 2 × 4 , and Res_{2×4}^{H_{y}}(τ_{2}) = t^{2} and Res_{2×4}^{H_{y}}(L_{1}) = t^{2}a. Finally, Res_{H_{by}}^{H_{y}}(L_{1}^{2} + w^{2}Z_{4} + a^{2}τ_{2}^{2} + w^{2}τ_{2}^{2}) = (Z_{2}w + at^{2} + atw + t^{2}w + tw^{2})^{2} + w^{2}(Z_{2}^{2}) + a^{2}(t^{4} + t^{2}w^{2}) + w^{2}(t^{4} + t^{2}w^{2}) = 0 .

Lemma 19 For H_{a} ⊂ 2^{2+4} , the restriction and transfer maps satisfy

T r(τ_{2}) = wy + y^{2} + by + ay
T r(Z_{2}) = w^{2} + wy + by + aw + ab + b^{2}
T r(γ_{2}) = w^{2} + wy + y^{2} + by + aw + ay
Res(L_{1}) = Z_{2}w + (τ_{2} + γ_{2})(w + y)
Res(L_{2}) = Z_{2}y + τ_{2}(w + b)
Res(T_{4}) = τ_{2}^{2} + τ_{2}(wy + y^{2} + by)
Res(Z_{4}) = Z_{2}^{2} + Z_{2}(w^{2} + wy).

Proof: First consider the three transfer formulas. Clearly the images are linearly independent, seen by restricting to ⟨w, y⟩. They are also in the kernel of the cup product with the cohomology class a. From the relations in H^{∗}(2^{2+4}; 𝔽_{2}) given in formulas (6), we obtain

a(wy + y^{2} + by + ay) = awy + ay^{2} + aby + (ab + b^{2} + aw + by + ay)y
= b^{2}y + by^{2} = 0,
a(w^{2} + wy + by + aw + ab + b^{2}) = aw^{2} + awy + aby + ab^{2} + (ab + b^{2} + aw + by + ay)(w + b)
= b^{3} + b^{2}y + wb^{2} + wby
= b^{3} + b^{2}y + aby + ay^{2} = 0,
a(w^{2} + wy + y^{2} + by + aw + ay) = aw^{2} + awy + ay^{2} + aby + a^{2}y + (ab + b^{2} + aw + by + ay)w
= ay^{2} + aby + a^{2}y + abw + b^{2}w + bwy = 0.

To verify the transfer formulas, it is sufficient to compute the restrictions to the subgroup 2^{4} . The double coset formula yields Res_{2^{4}}^{2^{2+4}}T r_{H_{a}}^{2^{2+4}} = (1 + c_{a})Res_{2^{4}}^{H_{a}} . The group commutators [w, a] = z and [a, y] = zt imply the actions on cohomology c_{a}(t) = t + y and c_{a}(z) = z + w + y. From the proof of Lemma 17, we need to compute

\begin{aligned}(1 + c_{a})(t^{2} + tw) & = wy + y^{2} \\ (1 + c_{a})(z^{2} + zy) & = w^{2} + wy \\ (1 + c_{a})(zw + ty) & = w^{2} + wy + y^{2}. \end{aligned}

We now turn to the restriction formulas. We have T r(τ_{2}^{2}) = (wy + y^{2} + by + ay)^{2} = w^{2}y^{2} + y^{4} + awy^{2} + by^{3} = T r((wy + y^{2})τ_{2}) , using the formulas (6). Also, T r(Z_{2}^{2}) = (w^{2} + wy + by + aw + ab + b^{2})^{2} = w^{4} + w^{2}y^{2} + w^{3}a + a^{2}y^{2} = T r((w^{2} + wy)Z_{2}) . Therefore τ_{2}^{2} + (wy + y^{2})τ_{2} and Z_{2}^{2} + (w^{2} + wy)Z_{2} are both in the image of the restriction map. Similarly, we compute T r(Z_{2}y + τ_{2}(w + b)) = 0 and T r(Z_{2}w + (τ_{2} + γ_{2})(w + y)) = 0 . By comparing with the filtered descriptions in the spectral sequence (1), we may choose the classes L_{2} and L_{1} with the appropriate restrictions. We also choose Z_{4} so that Res(Z_{4}) = Z_{2}^{2} + (w^{2} + wy)Z_{2} . Note that Res(bL_{1}) = byτ_{2} , since in Lemma 17 we have wb = 0 and bγ_{2} = 0 .

We choose the class T_{4} so as to agree with the inflation of T_{4} from the cohomology of Syl_{2}(GL_{4}(𝔽_{2})) . In other words, without using relations derived from the differentials in the spectral sequence (4), we have

\begin{aligned}T r(τ_{2}^{2}) & = w^{2}y^{2} + y^{4} + b^{2}y^{2} + ay^{3} + awy^{2} + aby^{2} \\ T r(wyτ_{2}) & = w^{2}y^{2} + wy^{3} + ay^{3} + awy^{2} \\ T r(y^{2}τ_{2}) & = wy^{3} + y^{4} + by^{3} + ay^{3} \\ T r(byτ_{2}) & = ay^{3} + by^{3} + b^{2}y^{2} + aby^{2}. \end{aligned}

Therefore we choose Res(T_{4}) = τ_{2}^{2} + (wy + y^{2} + by)τ_{2} .

Lemma 20 For H_{y} ⊂ 2^{2+4} the restriction and transfer maps satisfy

\begin{aligned}T r(τ_{2}) & = b^{2} + by \\ Res(L_{2}) & = (w + b)τ_{2} \\ Res(T_{4}) & = τ_{2}^{2} + b^{2}τ_{2}. \end{aligned}

Proof: The relation b^{2}y + by^{2} listed in formulas (9) implies that b^{2} + by is in the image of the transfer map, and the only possibility is T r(τ_{2}) . Then T r(τ_{2}^{2}) = (b^{2} + by)^{2} = b^{3}y + ab^{2}y + aby^{2} + b^{2}y^{2} = 0 , and T r(bτ_{2}) = b^{3} + b^{2}y = aby + ay^{2} = wb^{2} + wby = T r(wτ_{2}) . By comparing with lemma 19, we have Res(L_{2}) = (w + b)τ_{2} . To see that Res(T_{4}) = τ_{2}^{2} + b^{2}τ_{2} = τ_{2}^{2} + Res(bL_{2}) , restrict further to H_{wy} = 2^{2+4} ∩ 2_{−}^{1+4} . Theorem 10 implies Res_{H_{wy}}^{S_{J}}(T_{4}) = t^{4} , and the proof of lemma 18 shows that Res_{H_{wy}}^{H_{y}}(τ_{2}^{2} + b^{2}τ_{2}) = (t^{2} + tb)^{2} + b^{2}(t^{2} + tb) = t^{4} + tb^{3} = t^{4} .

Alternatively, the restriction maps would also follow from a discussion of the cohomology of the group T′_{2} , analogous to lemma 13 for the group T_{2} .

Lemma 21 The relations in the cohomology of 2^{2+4} include

\begin{aligned}abL_{1} & = ayL_{1} + (ab + ay)L_{2} \\ b^{2}L_{1} & = byL_{1} + (ab + ay)L_{2} \\ b^{2}L_{2} & = byL_{2} \\ abyL_{2} & = ay^{2}L_{2} \\ L_{1}^{2} & = w^{2}Z_{4} + (a^{2} + w^{2} + y^{2})T_{4} + (w^{2}y + wy^{2})L_{1} \\ L_{2}^{2} & = y^{2}Z_{4} + (b^{2} + w^{2})T_{4} + (w^{2}y + wy^{2})L_{2}. \end{aligned}

Proof: The subgroups H_{y} and H_{a} do not detect the cohomology of 2^{2+4} (in fact, I do not believe that any set of proper subgroups of 2^{2+4} detects its cohomology) but we begin by considering the two restrictions. First, the restriction map to H_{y} satisfies

Res((b^{2} + by)L_{2}) = b^{2}(w + b)τ_{2} = 0
Res((a^{2} + aw)L_{1}) = (ab + b^{2})L_{1} = (a + b)abτ_{2} = b^{3}τ_{2} = 0
Res((a^{2} + aw)L_{2} + (b^{2} + by)L_{1}) = (ab + b^{2})(w + b)τ_{2} + b^{2}L_{1} = ab^{2}τ_{2} + b(abτ_{2}) = 0
Res(L_{1}^{2} + w^{2}Z_{4} + (a^{2} + w^{2})T_{4}) = L_{1}^{2} + w^{2}Z_{4} + (a^{2} + w^{2})(τ_{2}^{2} + b^{2}τ_{2}) = 0
Res(L_{2}^{2} + (b^{2} + w^{2})T_{4}) = (w + b)^{2}τ_{2}^{2} + (b^{2} + w^{2})(τ_{2}^{2} + b^{2}τ_{2}) = 0.

The restriction map for H_{a} ⊂ 2^{2+4} satisfies

Res(a^{2} + aw) = 0
Res(b^{2} + by) = 0
Res(L_{1}^{2} + w^{2}Z_{4} + (w^{2} + y^{2})T_{4} + (w^{2}y + wy^{2})L_{1}) = (Z_{2}w + (w + y)(τ_{2} + γ_{2}))^{2}
+ w^{2}(Z_{2}^{2} + (w^{2} + wy)Z_{2})
+ (w^{2} + y^{2})(τ_{2}^{2} + (wy + y^{2} + by)τ_{2})
+ (w^{2}y + wy^{2})(Z_{2}w + (w + y)(τ_{2} + γ_{2}))
= (w^{2} + y^{2})γ_{2}^{2} + (w^{4} + w^{2}y^{2})Z_{2}
+ (w^{3}y + wy^{3})γ_{2}
+ (w^{2}y^{2} + y^{4} + by^{3})τ_{2} = 0
Res(L_{2}^{2} + y^{2}Z_{4} + (b^{2} + w^{2})T_{4} + (w^{2}y + wy^{2})L_{2}) = (Z_{2}y + (w + b)τ_{2})^{2} + y^{2}(Z_{2}^{2} + (w^{2} + wy)Z_{2})
+ (b^{2} + w^{2})(τ_{2}^{2} + (wy + y^{2} + by)τ_{2})
+ (w^{2}y + wy^{2})(Z_{2}y + (w + b)τ_{2}) = 0.

Let K denote the kernel of the map determined by the two restriction maps

H^{∗}(2^{2+4}; 𝔽_{2}) → H^{∗}(H_{a}; 𝔽_{2}) ⊕ H^{∗}(H_{y}; 𝔽_{2}).

Then

K = 𝔽_{2}[T_{4}, Z_{4}] ⊗ {aby + ay𝔽_{2}[w, y]} .

Let I denote the image of the restriction map Res_{2^{2+4}}^{S_{J}} . Then

K ∩ I = wβ𝔽_{2}[T_{4}, Z_{8}]

where wβ denotes aby + ay^{2} , and Z_{8} denotes Z_{4}^{2} + Z_{4}T_{4} . Note that every element of K ∩ I has dimension congruent to 3 modulo 4 .

In spectral sequence (4) for S_{J} , we find filtered relations involving αL_{1} , βL_{2} , αL_{2} +βL_{1} , L_{1}^{2} + wM_{1} , and L_{2}^{2} + yM_{2} . These relations restrict to corresponding relations in the cohomology of 2^{2+4} , which must be completely detected on the two subgroups H_{a} and H_{y} since they occur in dimensions 5 and 6 . We have thus proven, in H^{∗}(2^{2+4}; 𝔽_{2}) ,

\begin{aligned}(a^{2} + aw)L_{1} & = 0 \\ (b^{2} + by)L_{2} & = 0 \\ (a^{2} + aw)L_{2} & = (b^{2} + by)L_{1} \end{aligned}  (9)

as well as the two relations involving L_{1}^{2} and L_{2}^{2} in lemma 2.1.

To complete the proof, note that ab+ay = a^{2}+aw+b^{2}+by, so that (ab+ay)L_{1} = (b^{2}+by)L_{1} and (ab + ay)L_{2} = (a^{2} + aw)L_{2} . Finally, (aby + ay^{2})L_{2} = w(b^{2} + by)L_{2} = 0 .

In spectral sequence (1) for 2^{2+4} , there are many filtered relations involving the class R_{2} . To determine the actual relations, we must more precisely define the class R_{2} . We will use the transfer map from the subgroup H_{a} to do this.

Lemma 22 For H_{a} ⊂ 2^{2+4} we have transfers

\begin{aligned}T r(τ_{2}Z_{2}) & = (w + y + a + b)L_{2} + w^{3}y + wy^{3} + aw^{2}y + ay^{3} \\ T r(τ_{2}γ_{2}) & = (w + a)L_{2} + (y + b)L_{1} + by^{3} + w^{3}y + y^{4} + aw^{2}y \\ T r(γ_{2}Z_{2}) & = (w + a)L_{1} + (w + y + a + b)L_{2} + aw^{3} + w^{4} + wy^{3} + ay^{3}. \end{aligned}

Proof: Applying lemma 19, the double coset formula, and the relations from lemma 17, we obtain

Res ∘ T r(τ_{2}Z_{2}) = (w^{2} + wy)τ_{2} + (wy + y^{2} + by)Z_{2} + w^{3}y + wy^{3}
= Res((w + y + b)L_{2} + w^{3}y + wy^{3})
Res ∘ T r(τ_{2}γ_{2}) = (w^{2} + wy + y^{2} + by)τ_{2} + (wy + y^{2} + by)γ_{2} + by^{3} + w^{3}y + y^{4}
= Res(wL_{2} + (b + y)L_{1} + by^{3} + w^{3}y + y^{4})
Res ∘ T r(γ_{2}Z_{2}) = (w^{2} + wy)γ_{2} + (w^{2} + wy + y^{2} + by)Z_{2} + w^{4} + wy^{3}
= Res(wL_{1} + (w + y + b)L_{2} + w^{4} + wy^{3}).

The double coset formula also yields Res_{H_{y}}^{2^{2+4}}T r_{H_{a}}^{2^{2+4}} = T r_{H_{ay}}^{H_{y}}Res_{H_{ay}}^{H_{a}} . In particular,

\begin{aligned}T r_{H_{ay}}^{H_{y}}(γ_{2}) & = w^{2} + aw \\ T r_{H_{ay}}^{H_{y}}(Z_{2}) & = w^{2} + aw + ab + b^{2}. \end{aligned}

Therefore

Res_{H_{y}}^{2^{2+4}}T r_{H_{a}}^{2^{2+4}}(τ_{2}Z_{2}) = (w^{2} + aw + ab + b^{2})τ_{2}

= Res_{H_{y}}^{2^{2+4}}((w + a + b)L_{2})

Res_{H_{y}}^{2^{2+4}}T r_{H_{a}}^{2^{2+4}}(τ_{2}γ_{2}) = (w^{2} + aw)τ_{2}

= Res_{H_{y}}^{2^{2+4}}((w + a)L_{2} + bL_{1}).

Recall that the cohomology of H_{y} is detected on the two subgroups H_{wy} and H_{by} . We also use the subgroups 2 × 4 = ⟨z, t, b⟩ and 2^{3} = ⟨z, t, w⟩. Another application of the double coset formula yields

Res_{H_{wy}}^{H_{y}}T r_{H_{ay}}^{H_{y}}(γ_{2}Z_{2}) = T r_{2×4}^{H_{wy}}Res_{2×4}^{H_{ay}}(γ_{2}Z_{2})
= T r_{2×4}^{H^{wy}}(btZ_{2})
= a^{2}bt
= t^{2}(a^{2} + ab + b^{2}) + t(ab^{2})
= Res_{H_{wy}}^{H_{y}}(aL_{1} + (ab + b^{2})τ_{2})
Res_{H_{by}}^{H_{y}}T r_{H_{ay}}^{H_{y}}(γ_{2}Z_{2}) = T r_{2^{3}}^{H_{by}}Res_{2^{3}}^{H_{ay}}(γ_{2}Z_{2})
= T r_{2^{3}}^{H_{by}}(z^{3}w)
= (w^{2} + aw)Z_{2} + w^{4} + aw^{3}
= Res_{H_{wy}}^{H_{y}}((w + a)L_{1} + (w^{2} + wa)τ_{2} + w^{4} + aw^{3}).

Therefore

Res_{H_{y}}^{2^{2+4}}T r_{H_{a}}^{2^{2+4}}(γ_{2}Z_{2}) = T r_{H_{ay}}^{H_{y}}(γ_{2}Z_{2})
= (w + a)L_{1} + (w^{2} + wa + ab + b^{2})τ_{2} + w^{4} + aw^{3}
= Res_{H_{y}}^{2^{2+4}}((w + a)L_{1} + (w + a + b)L_{2} + w^{4} + aw^{3}).

We have now determined the transfer formulas up to multiples of ay, that is classes in K . In dimension 4 there are eight possibilities, in the span ay{w^{2} + wy + y^{2}}. The image of the transfer map is equal to the kernel of the cup product with the class a, so we compare with the relations described in lemma 21. By using formulas (6), which imply that a^{2}w^{2}y = aw^{3}y, a^{2}wy^{2} = aw^{2}y^{2} , and a^{2}y^{3} = awy^{3} , the result is proven.

Let us denote by U the class T r_{H_{a}}^{2^{2+4}}(τ_{2}γ_{2}Z_{2}) . Then Res_{H_{y}}^{2^{2+4}}(U) = T r_{H_{ay}}^{H_{y}}(τ_{2}γ_{2}Z_{2}) = τ_{2}((w+ a)L_{1} +(a^{2} +w^{2})τ_{2} +aw^{3} +w^{4}) , using lemma 22. Note that aτ_{2}L_{1} is the essential part, detecting some of the filtered description R_{2} = z^{2}t^{2}(aw + by + ay) from spectral sequence (1) . We also have Res_{H_{a}}^{2^{2+4}}(U) = Z_{2}τ_{2}(w^{2} + wy + y^{2} + by) + Z_{2}γ_{2}(wy + y^{2}) + τ_{2}γ_{2}(w^{2} + wy) + Z_{2}(w^{3}y + y^{4} + by^{3}) + τ_{2}(w^{4} + wy^{3}) + γ_{2}(w^{3}y + wy^{3}) + w^{5}y + w^{4}y^{2} + w^{2}y^{4} + wy^{5} . The essential part in the restriction is byτ_{2}Z_{2} . We might have defined R_{2} by the two restrictions aτ_{2}L_{1} and byτ_{2}Z_{2} , but this would only determine R_{2} up to a multiple of ay. Instead, we choose a different definition of R_{2} as

R_{2} = c_{x}(U).  (10)

Lemma 23 The class U = T r_{Ha}^{2^{2+4}}(τ_{2}γ_{2}Z_{2}) satisfies the relations

aU = 0
bU = 0
wU = (w + a)L_{1}L_{2} + (w^{2}y + awy)Z_{4} + (w^{3} + w^{2}y + ab^{2} + aw^{2} + awy + ay^{2})T_{4}

+ (w^{3}y + w^{2}y^{2} + aw^{2}y + awy^{2})L_{1} + (w^{4} + wy^{3} + aw^{3} + ay^{3})L_{2} + (w + a)(w^{5}y + w^{4}y^{2} + w^{2}y^{4} + wy^{5})

yU = (y + b)L_{1}L_{2} + (wy^{2} + ay^{2})Z_{4} + (w^{2}y + wy^{2} + ab^{2} + aby + awy + ay^{2})T_{4}

+ (w^{2}y^{2} + wy^{3} + awy^{2} + ay^{3})L_{1} + (w^{3}y + y^{4} + aw^{2}y + by^{3})L_{2} + (w + a)(w^{4}y^{2} + w^{3}y^{3} + wy^{5} + y^{6})

L_{1}U = (wy + ay)L_{1}Z_{4} + (w^{2} + aw)L_{2}Z_{4} + (w^{2} + wy + aw + ay)L_{1}T_{4}

+ (w^{2} + y^{2} + aw + by)L_{2}T_{4} + (w^{4}y + w^{3}y^{2} + aw^{2}y^{2} + aw^{3}y)Z_{4} + (w^{4}y + w^{3}y^{2} + w^{2}y^{3} + wy^{4} + aw^{3}y + aw^{2}y^{2} + awy^{3} + ay^{4})T_{4} + (w^{3} + w^{2}y + wy^{2} + y^{3} + aw^{2} + awy + ay^{2} + by^{2})L_{1}L_{2} + (w^{5}y + wy^{5} + aw^{4}y + ay^{5})L_{1}

L_{2}U = (y^{2} + by)L_{1}Z_{4} + (wy + ay)L_{2}Z_{4} + (w^{2} + aw)L_{1}T_{4}

+ (w^{2} + wy + aw + ay)L_{2}T_{4} + (w^{3}y^{2} + y^{5} + aw^{2}y^{2} + by^{4})Z_{4} + (w^{5} + w^{2}y^{3} + aw^{4} + awy^{3})T_{4}

U^{2} = Z_{4}L_{1}(w^{2}y^{3} + wy^{4} + awy^{3} + ay^{4}) + Z_{4}L_{2}(w^{4}y + w^{3}y^{2} + aw^{3}y + aw^{2}y^{2})

+ Z_{4}T_{4}(w^{4} + w^{2}y^{2} + y^{4} + aw^{3} + awy^{2} + by^{3}) + Z_{4}(w^{4}y^{4} + y^{8} + aw^{3}y^{4} + by^{7}) + L_{1}L_{2}(w^{4}y^{2} + w^{2}y^{4} + aw^{3}y^{2} + awy^{4}) + T_{4}L_{1}(w^{4}y + w^{3}y^{2} + aw^{3}y + aw^{2}y^{2}) + T_{4}L_{2}(w^{4}y + w^{3}y^{2} + w^{2}y^{3} + wy^{4} + aw^{3}y + aw^{2}y^{2} + awy^{3} + ay^{4}) + L_{1}(w^{6}y^{3} + w^{5}y^{4} + w^{4}y^{5} + w^{3}y^{6} + aw^{5}y^{3} + aw^{4}y^{4} + aw^{3}y^{5} + aw^{2}y^{6}) + L_{2}(w^{8}y + w^{7}y^{2} + w^{2}y^{7} + wy^{8} + aw^{7}y + aw^{6}y^{2} + awy^{7} + ay^{8}) + T_{4}(w^{8} + w^{6}y^{2} + aw^{5}y^{2} + aw^{7}) + (w + a)(w^{9}y^{2} + w^{7}y^{4} + w^{3}y^{8} + wy^{10}).

Proof: The relation aU = 0 follows immediately from the definition of U and the Gysin sequence for H_{a} ⊂ 2^{2+4} . The rest of the relations follow from the formula T r(A ∪ Res(B)) = T r(A) ∪ B . For example, bU = bT r(τ_{2}γ_{2}Z_{2}) = T r(bτ_{2}γ_{2}Z_{2}) = 0 , since bγ_{2} = 0 was shown in lemma 17. As another example,

L_{2}U = L_{2}T r(τ_{2}γ_{2}Z_{2})
= T r((Z_{2}y + (w + b)τ_{2})τ_{2}γ_{2}Z_{2})
= T r((Z_{2}^{2} + (w^{2} + wy)Z_{2})yτ_{2}γ_{2} + (τ_{2}^{2} + (wy + y^{2} + by)τ_{2})wγ_{2}Z_{2})
= Z_{4}yT r(τ_{2}γ_{2}) + T_{r}wT r(γ_{2}Z_{2})
= Z_{4}y((w + a)L_{2} + (y + b)L_{1} + by^{3} + w^{3}y + y^{4} + aw^{2}y)
+ T_{4}w((w + a)L_{1} + (w + y + a + b)L_{2} + aw^{3} + w^{4} + wy^{3} + ay^{3}),

which is equivalent to the relation given. We have used lemmas 19 and 22 to evaluate transfers. The other relations follow similarly.

Theorem 24 The class R_{2} ∈ H^{6}(2^{2+4}; 𝔽_{2}) defined by R_{2} = c_{x}(T r_{H_{a}}^{2^{2+4}}(τ_{2}γ_{2}Z_{2})) satisfies the relations

wR_{2} = aR_{2} = aL_{1}L_{2} + awyZ_{4} + (aw^{2}y + awy^{2})L_{1} + (aw^{3} + ay^{3})L_{2}
+ (ab^{2} + aby + aw^{2})T_{4} + aw^{5}y + aw^{4}y^{2} + aw^{2}y^{4} + awy^{5}
yR_{2} = bR_{2} = bL_{1}L_{2} + ay^{2}Z_{4} + (awy^{2} + ay^{3})L_{1} + (aw^{2}y + by^{3})L_{2}
+ (ab^{2} + awy + ay^{2})T_{4} + aw^{4}y^{2} + aw^{3}y^{3} + awy^{5} + ay^{6}
L_{1}R_{2} = ayL_{1}Z_{4} + awL_{2}Z_{4} + awL_{1}T_{4} + byL_{2}T_{4} + (aw^{2}y^{2} + aw^{3}y)Z_{4}
+ (awy^{3} + ay^{4})T_{4} + (aw^{2} + awy + ay^{2} + by^{2})L_{1}L_{2} + (aw^{4}y + ay^{5})L_{1}
L_{2}R_{2} = byL_{1}Z_{4} + ayL_{2}Z_{4} + (aw + by)L_{1}T_{4} + awL_{2}T_{4}
+ (aw^{2}y^{2} + by^{4})Z_{4} + (aw^{4} + aw^{2}y^{2} + awy^{3} + by^{4})T_{4}
R_{2}^{2} = (awy^{3} + ay^{4})L_{1}Z_{4} + (aw^{3}y + aw^{2}y^{2})L_{2}Z_{4} + (aw^{3} + awy^{2} + by^{3})Z_{4}T_{4}
+ (aw^{3}y^{4} + by^{7})Z_{4} + (aw^{3}y^{2} + awy^{4})L_{1}L_{2} + (aw^{3}y + aw^{2}y^{2} + awy^{3} + ay^{4})L_{1}T_{4}
+ (awy^{3} + ay^{4})L_{2}T_{4} + (aw^{5}y^{3} + aw^{4}y^{4} + aw^{3}y^{5} + aw^{2}y^{6})L_{1}
+ (aw^{7}y + aw^{6}y^{2} + awy^{7} + ay^{8})L_{2} + (aw^{3} + awy^{2} + by^{3})T_{4}^{2}
+ (aw^{7} + aw^{5}y^{2} + aw^{3}y^{4} + by^{7})T_{4} + aw^{9}y^{2} + aw^{7}y^{4} + aw^{3}y^{8} + awy^{10}.

Proof: This is an immediate consequence of lemma 23 and the definition R_{2} = c_{x}(U) . Note that the conjugation map c_{x} fixes the classes w, y, L_{1} , L_{2} , and T_{4} , since these classes restrict from the cohomology of 2^{2+4} . We also have c_{x}(a) = a+w, c_{x}(b) = b+y, and c_{x}(Z_{4}) = Z_{4} +T_{4} . Thus aR_{2} = ac_{x}(U) = c_{x}(a + w)U = c_{x}(wU) , using aU = 0 . Also wR_{2} = wc_{x}(U) = c_{x}(wU) . Then compute the map c_{x} on the relation given for wU in lemma 23. The calculations are straight forward.

Lemma 25 The class R_{1} = T r_{2^{2+4}}^{S_{J}}(U) can be written as

\begin{aligned}R_{1} & = L_{1}L_{2} + yM_{1} + (w^{2} + wy + α + β)T_{4} + (w^{2}y + wy^{2})L_{1} \\ & + (w^{3} + y^{3})L_{2} + w^{5}y + w^{4}y^{2} + w^{2}y^{4} + wy^{5}. \end{aligned}

Proof: The cohomology of S_{J} is detected by the two subgroups 2_{−}^{1+4} and 2^{2+4} , so we restrict the given equation, utilizing the double coset formula. We have

Res_{2_{−}^{1+4}}^{S_{J}}(R_{1}) = T r_{H_{wy}}^{2_{−}^{1+4}}Res_{H_{wy}}^{2^{2+4}}(U)
= T r_{H_{wy}}^{2_{−}^{1+4}}T r_{H_{awy}}^{H_{wy}}Res_{H_{awy}}^{H_{a}}(τ_{2}γ_{2}Z_{2}), where H_{awy} ≃ 2 × 4,
= T r_{H_{wy}}^{2_{−}^{1+4}}T r_{H_{awy}}^{H_{wy}}(t^{3}bZ_{2})
= T r_{H_{wy}}^{2_{−}^{1+4}}(t^{3}a^{2}b)
= 0.

We apply Theorem 10 and the relations from lemma 1 to show that the right-hand-side of the given equation restricts to zero.

Res_{2_{−}^{1+4}}^{S_{J}}(L_{1}L_{2} + (α + β)T_{4}) = (t^{2}a + tab + tb^{2} + tbx)(t^{2}b + t^{2}x + tb^{2} + tbx)
+ (a^{2} + ax + b^{2} + bx)(t^{4} + t^{3}x)
= t^{4}(a^{2} + ab + b^{2} + bx) + t^{3}(b^{3} + a^{2}x + ax^{2} + b^{2}x)
+ t^{2}(b^{4} + ab^{3} + ab^{2}x + b^{2}x^{2})
= t^{4}(tx + ax) + t^{3}(t^{2}x + tbx + tx^{2} + abx + b^{2}x + bx^{2})
+ t^{2}(t^{2}ax + t^{2}bx + t^{2}x^{2} + tb^{2}x + tbx^{2} + tabx)
= 0.

Since the dimension of R_{1} is not congruent to 3 modulo 4 , the restriction to 2^{2+4} is detected on the two subgroups H_{a} and H_{y} . The double coset formula implies that

Res_{2^{2+4}}^{S_{J}}(R_{1}) = (1 + c_{x})U.

Therefore we restrict the classes U and R_{2} = c_{x}(U) to both H_{a} and H_{y} .

Res_{H_{y}}^{2^{2+4}}(U) = Res_{H_{y}}^{2^{2+4}}T r_{H_{a}}^{2^{2+4}}(τ_{2}γ_{2}Z_{2})
= T r_{H_{ay}}^{H_{y}}Res_{H_{ay}}^{H_{a}}(τ_{2}γ_{2}Z_{2})
= T r_{H_{ay}}^{H_{y}}(τ_{2}γ_{2}Z_{2})
= τ_{2}T r_{H_{ay}}^{H_{y}}(γ_{2}Z_{2})
= τ_{2}Res_{H_{y}}^{2^{2+4}}T r_{H_{a}}^{2^{2+4}}(γ_{2}Z_{2})
= τ_{2}((w + a)L_{1} + (w + a + b)(w + b)τ_{2} + aw^{3} + w^{4}).
Res_{H_{y}}^{2^{2+4}} = Res_{H_{y}}^{2^{2+4}}(c_{x}(U))
= c_{x}Res_{H_{y}}^{2^{2+4}}(U)
= τ_{2}(aL_{1} + (a + b)(w + b)τ_{2} + aw^{3}).

Therefore

Res_{H_{y}}^{2^{2+4}}(R_{1}) = τ_{2}(wL_{1} + w^{2}τ_{2} + w^{4}).

The restriction of the right-hand-side of the given equation is

Res_{H_{y}}^{2^{2+4}}(L_{1}L_{2} + (w^{2} + a^{2} + aw + b^{2})T_{4} + w^{3}L_{2}) = L_{1}(w + b)τ_{2} + (w^{2} + ab)(τ_{2}^{2} + b^{2}τ_{2})
+ w^{3}(w + b)τ_{2}
= τ_{2}(wL_{1} + abτ_{2} + (w^{2} + ab)τ_{2} + w^{4}).

The restriction to the group H_{a} of the class U is described by the double coset formula as

Res_{H_{a}}^{2^{2+4}}(U) = (1 + c_{a})(τ_{2}γ_{2}Z_{2}).

To compute the action of the conjugation map c_{a} , first restrict to the subgroup H_{ab} = 2^{4} . The commutators of the group elements [a, w] = z and [a, y] = zt imply that the conjugation action on the cohomology of H_{ab} is c_{a}(z) = z + w + y, c_{a}(t) = t + y, and c_{a} fixes w and y. Therefore

\begin{aligned}c_{a}(t^{2} + tw) & = (t + y)^{2} + (t + y)w = t^{2} + tw + wy + y^{2} \\ c_{a}(zw + ty) & = (z + w + y)w + (t + y)y = zw + ty + w^{2} + wy + y^{2} \\ c_{a}(z^{2} + zy) & = (z + w + y)^{2} + (z + w + y)y = z^{2} + zy + w^{2} + wy. \end{aligned}

This determines the action of c_{a} on the classes τ_{2} , γ_{2} , and Z_{2} up to a multiple of b, which in dimension two means the single class by. This can be detected on the elementary abelian group 2^{3} generated by the group elements ⟨z, t, by⟩. Denote the cohomology class dual to the group element by as \overline{y}. Then Res_{2^{3}}^{H_{a}}(b) = Res_{2^{3}}^{H_{a}}(y) = \overline{y}, and the action of c_{a} on the cohomology of 2^{3} is c_{a}(t) = t + \overline{y}. Therefore

\begin{aligned}c_{a}(t^{2} + t\overline{y}) & = (t + \overline{y})^{2} + (t + \overline{y})\overline{y} = t^{2} + t\overline{y} \\ c_{a}(z^{2} + z\overline{y}) & = z^{2} + z\overline{y}. \end{aligned}

Since Res_{H_{aw}}^{H_{a}}(τ_{2}) = t^{2} + tb, and Res_{H_{aw}}^{H_{a}}(γ_{2}) = tb + ty, combining the above results implies that c_{a} acts on the cohomology of H_{a} as

\begin{aligned}c_{a}(τ_{2}) & = τ_{2} + wy + y^{2} + by \\ c_{a}(γ_{2}) & = γ_{2} + w^{2} + wy + y^{2} + by \\ c_{a}(Z_{2}) & = Z_{2} + w^{2} + wy. \end{aligned}

Thus

Res_{H_{a}}^{2^{2+4}}(U) = τ_{2}γ_{2}Z_{2} + (τ_{2} + wy + y^{2} + by)(γ_{2} + w^{2} + wy + y^{2} + by)(Z_{2} + w^{2} + wy).

We briefly utilize the group H_{a}^{x} = x(H_{a})x^{−1} , the subgroup conjugate to the group H_{a} . Then

Res_{H_{a}}^{2^{2+4}}(c_{x}U) = c_{x}Res_{H_{a}^{x}}^{2^{2+4}}(U)
= c_{x}T r_{H_{aw}}^{H_{a}^{x}}Res_{H_{aw}}^{H_{a}}(τ_{2}γ_{2}Z_{2})
= T r_{H_{aw}}^{H_{a}}c_{x}Res_{H_{aw}}^{H_{a}}(τ_{2}γ_{2}Z_{2})
= T r_{H_{aw}}^{H_{a}}c_{x}(t^{2} + tb)(tb + ty)Z_{2}
= T r_{H_{aw}}^{H_{a}}c_{x}(t^{3})(b + y)Z_{2}.

The action of the conjugation map c_{x} on the group H_{aw} ≃ 2 × D_{8} can be determined by restricting to the two elementary abelian groups 2^{3} = ⟨z, t, y⟩ and 2_{x}^{3} = ⟨z, t, by⟩. Denote the cohomology element dual to the group element by as \overline{y}. Then c_{x}(t) = t, c_{x}(z) = z + t, and c_{x}(y) = \overline{y}. We obtain the action of c_{x} on H_{aw} as c_{x}(t) = t, c_{x}(y) = y, c_{x}(b) = b + y, and c_{x}(Z_{2}) = Z_{2} + t^{2} + ty. Therefore

Res_{H_{a}}^{2^{2+4}}(c_{x}U) = T r_{H_{aw}}^{H_{a}}(t^{3}b)(Z_{2} + t^{2} + ty)

= T r_{H_{aw}}^{H_{a}}(tb(t^{2} + tb)Z_{2} + t^{2}b^{2}Z_{2} + tb(t^{2} + tb)^{2} + by(t^{2} + tb)^{2} + tby^{2}(t^{2} + tb))

= byτ_{2}Z_{2} + by^{3}Z_{2} + byτ_{2}^{2} + by^{3}τ_{2},

where we have used T r(t) = b, Res(τ_{2}) = t^{2}+tb, and the relation b^{2} = by. This is the restriction of the class R_{2} , and the restriction of R_{1} immediately follows. I will skip the final computation, which uses the above along with lemma 19 to verify the given equation for R_{1} when restricted to H_{a} .

We now turn to a study of the class V ∈ H^{10}(S_{J}; 𝔽_{2}) . This class has been determined up to a multiple of x by the restriction Res_{2^{2+4}}^{S_{J}}(V) = Z_{4}R_{1} + T_{4}R_{2} given in Theorem 8. Let us restrict further to the subgroup H_{y} .

Lemma 26 Res_{H_{y}}^{S_{J}}(V) = Z_{4}(wτ_{2}L_{1} + w^{2}τ_{2}^{2} + w^{4}τ_{2}) + aτ_{2}^{3}L_{1} + (aw + ab + b^{2})τ_{2}^{4} + aw^{3}τ_{2}^{3} .

Proof: This follows immediately from the proof of lemma 25, which includes the restrictions

\begin{aligned}Res_{H_{y}}^{2^{2+4}}(R_{1}) & = wτ_{2}L_{1} + w^{2}τ_{2}^{2} + w^{4}τ_{2}^{2} \\ Res_{H_{y}}^{2^{2+4}}(R_{2}) & = aτ_{2}L_{1} + (aw + ab + b^{2})τ_{2}^{2} + aw^{3}τ_{2}. \end{aligned}

Also, lemma 20 states that Res_{H_{y}}^{2^{2+4}}(T_{4}) = τ_{2}^{2} + b^{2}τ_{2} , and lemma 18 gives the relations in the cohomology of H_{y} .

Let T′_{2} be the index two subgroup of S_{J} generated by the group elements ⟨z, t, a, b, w, x⟩. Note that T′_{2} is conjugate to the group T_{2} described in lemmas 12 and 13, and that T′_{2} ∩ 2^{2+4} = H_{y} .

Lemma 27 Res_{T′_{2}}^{S_{j}}(V) = τ_{2}L_{1}M_{1} + w^{2}L_{2}M_{1} + wM_{1}(τ_{2}^{2} + b^{2}τ_{2}) + α(τ_{2}^{4} + b^{4}τ_{2}^{2} + b^{2}x^{2}τ_{2}^{2}) .

Proof: A study of the Gysin sequence for T′_{2} ⊂ S_{J} yields results analogous to those for T_{2} given in lemma 13. In particular, we have T r_{T′_{2}}^{S_{J}} (τ_{2}) = β and Res_{T′_{2}}^{S_{J}} (T_{4}) = τ_{2}^{2} + (b^{2} + bx)τ_{2} . Also, Res_{T′_{2}}^{S_{J}} (β) = b^{2} + bx.

I claim that the following relations hold in the cohomology of S_{J} :

\begin{aligned}βL_{1} & = αL_{2} \\ βL_{2} & = 0 \\ αM_{1} & = wβT_{4}. \end{aligned}  (11)

These relations can be detected by restricting to the two subgroups 2^{2+4} and 2_{−}^{1+4} . We therefore obtain

T r_{T′_{2}}^{S_{J}} (τ_{2}L_{1}M_{1}) = βL_{1}M_{1}
= αL_{2}M_{1}
= wβL_{2}T_{4}
= 0.

The Gysin sequence implies that τ_{2}L_{1}M_{1} is in the image of the restriction map. By comparing with the restrictions of other classes, this restriction must involve the class V . That is, τ_{2}L_{1}M_{1}+ Res(V) is in the image of the restriction map from S_{J} . Note that the given formula is equivalent to

τ_{2}L_{1}M_{1} + Res_{T′_{2}}^{S_{J}} (V) = Res_{T′_{2}}^{S_{J}} (w^{2}L_{2}M_{1} + wM_{1}T_{4} + αT_{4}^{2}).

To verify this formula, restrict to H_{y} = T′_{2} ∩2^{2+4} and compare with lemma 26. The result follows from the restrictions, derived from Theorem 8 and lemma 20.

Res_{H_{y}}^{S_{J}}(L_{2}) = (w + b)τ_{2}
Res_{H_{y}}^{S_{J}}(M_{1}) = wZ_{4} + a(τ_{2}^{2} + b^{2}τ_{2})
Res_{H_{y}}^{S_{J}}(T_{4}) = bτ_{2}^{2}.

Along with the relations wb = 0 , b^{3} = 0 , bL_{1} = abτ_{2} , and a^{2} = ab + b^{2} + aw from lemma 18, we obtain

Res_{H_{y}}^{2^{2+4}}(Z_{4}R_{1} + T_{4}R_{2}) = Res_{H_{y}}^{T′_{2}}(τ_{2}L_{1}M_{1}) + Res_{H_{y}}^{S_{J}}(w^{2}L_{2}M_{1} + wM_{1}T_{4} + αT_{4}^{2}).

Lemma 28 Res_{2_{−}^{1+4}}^{S_{J}}(V) = t^{7}ab^{2} + t^{3}ab^{2}x^{4} .

Proof: Since 2_{−}^{1+4} ⊂ T′_{2} , this is a corollary to lemma 27. We use the restrictions, given in Theorem 10,

Res_{2_{−}^{1+4}}^{S_{J}}(L_{1}) = (t^{2} + ta)(a + x)
Res_{2_{−}^{1+4}}^{S_{J}}(L_{2}) = (t^{2} + tb)(b + x)
Res_{2_{−}^{1+4}}^{S_{J}}(M_{1}) = a(t^{4} + t^{3}x)
Res_{2_{−}^{1+4}}^{S_{J}}(T_{4}) = t^{4} + t^{3}x.

We also use the formula, analogous to one given in the proof of Theorem 10,

Res_{2_{−}^{1+4}}^{T′_{2}}(τ_{2}) = t^{2} + tb.

Applying the relations given in lemma 1, we obtain

Res_{2_{−}^{1+4}}^{S_{J}}(V) = (t^{2} + tb)(t^{2} + ta)(a + x)(a)(t^{4} + t^{3}x)
+ (a^{2} + ax)(t^{4} + t^{3}x)^{2}
= t^{7}ab^{2} + t^{3}ab^{2}x^{4}.

Theorem 29 The cohomology of S_{J} , generated by classes {w, x, y, α, β, L_{1}, L_{2}, T_{4}, M_{1}, M_{2}, Z_{8}, V }, has relations spanned by the following:

(a) Relations detected on the bottom row E_{∞}^{∗,0} of spectral sequence (4), either inflated from the cohomology of Syl_{2}(GL_{4}(𝔽_{2})) or given by a differential to the bottom row:

\begin{aligned}wx & = 0 \\ wα & = 0 \\ α^{3} & = 0 \end{aligned}  \begin{aligned}xy & = 0 \\ yβ & = 0 \\ β^{2} & = α^{2} + αβ \end{aligned}  \begin{aligned}wβ & = yα \\ xT_{4} & = 0 \end{aligned}

(b) Relations detected as filtered relations in spectral sequence (4), appearing in the rows E_{∞}^{∗,2} , E_{∞}^{∗,4} , and E_{∞}^{∗,7} :

\begin{aligned}xL_{1} & = α^{2} \\ xM_{1} & = 0 \\ xV & = 0 \\ αL_{2} & = βL_{1} \\ βM_{2} & = wβT_{4} \end{aligned}  \begin{aligned}xL_{2} & = β^{2} \\ xM_{2} & = 0 \\ αV & = 0 \\ wM_{2} & = yM_{1} \\ αM_{2} & = βM_{1} + wβT_{4} \end{aligned}  \begin{aligned}αL_{1} & = 0 \\ βL_{2} & = 0 \\ βV & = 0 \\ αM_{1} & = wβT_{4} \end{aligned}

(c) Relations involving squares and products of the internal generators in spectral sequence (4):

L_{1}^{2} = wM_{1} + (α + w^{2} + y^{2})T_{4} + (w^{2}y + wy^{2})L_{1}
L_{2}^{2} = yM_{2} + (β + w^{2})T_{4} + (w^{2}y + wy^{2})L_{2}
M_{1}^{2} = w^{2}Z_{8} + wM_{1}T_{4} + αT_{4}^{2}
M_{2}^{2} = y^{2}Z_{8} + yM_{2}T_{4} + βT_{4}^{2}
M_{1}M_{2} = wyZ_{8} + yM_{1}T_{4} + (α + β)T_{4}^{2}
L_{1}V = (wyL_{1} + w^{2}L_{2} + w^{4}y + w^{3}y^{2})Z_{8} + (yL_{2}T_{4} + y^{2}L_{1}L_{2})M_{2}

+ (wL_{1}T_{4} + w^{2}L_{1}L_{2} + wyL_{1}L_{2} + y^{2}L_{1}L_{2} + w^{4}yL_{1} + y^{5}L_{1} + wy^{3}T_{4} + y^{4}T_{4})M_{1}

L_{2}V = (y^{2}L_{1} + wyL_{2} + w^{3}y^{2} + y^{5})Z_{8} + (yL_{1}T_{4} + y^{4}T_{4})M_{2}

+ (wL_{1}T_{4} + wL_{2}T_{4} + w^{4}T_{4} + w^{2}y^{2}T_{4} + wy^{3}T_{4})M_{1}

M_{1}V = (wZ_{8} + T_{4}M_{1})(L_{1}L_{2} + yM_{1} + (w^{2} + wy + α + β)T_{4}

+ (w^{2}y + wy^{2})L_{1} + (w^{3} + y^{3})L_{2} + w^{5}y + w^{4}y^{2} + w^{2}y^{4} + wy^{5})

M_{2}V = (yZ_{8} + T_{4}M_{2})(L_{1}L_{2} + yM_{1} + (w^{2} + wy + α + β)T_{4}

+ (w^{2}y + wy^{2})L_{1} + (w^{3} + y^{3})L_{2} + w^{5}y + w^{4}y^{2} + w^{2}y^{4} + wy^{5})

V^{2} = (w^{3} + wy^{2})T_{4}M_{1}Z_{8} + (wy^{3} + y^{4})L_{1}M_{1}Z_{8} + (w^{3}y + w^{2}y^{2})L_{2}M_{1}Z_{8} + w^{3}y^{4}M_{1}Z_{8}

+ y^{3}T_{4}M_{2}Z_{8} + y^{7}M_{2}Z_{8} + (w^{4} + w^{2}y^{2} + y^{4})T_{4}^{2}Z_{8} + (w^{4}y + w^{3}y^{2} + w^{2}y^{3} + wy^{4})L_{1}T_{4}Z_{8} + (w^{2}y^{3} + wy^{4})L_{2}T_{4}Z_{8} + (w^{8} + w^{6}y^{2} + w^{4}y^{4} + y^{8})T_{4}Z_{8} + (w^{4}y^{2} + w^{2}y^{4})L_{1}L_{2}Z_{8} + (w^{6}y^{3} + w^{5}y^{4} + w^{4}y^{5} + w^{3}y^{6})L_{1}Z_{8} + (w^{8}y + w^{7}y^{2} + w^{2}y^{7} + wy^{8})L_{2}Z_{8} + (w^{10}y^{2} + w^{8}y^{4} + w^{4}y^{8} + w^{2}y^{10})Z_{8} + (w^{3} + wy^{2})T_{4}^{3}M_{1} + y^{3}T_{4}^{3}M_{2} + (w^{3}y + w^{2}y^{2} + wy^{3} + y^{4})L_{1}T_{4}^{2}M_{1} + (wy^{3} + y^{4})L_{2}T_{4}^{2}M_{1} + y^{7}T_{4}^{2}M_{2} + (w^{7} + w^{5}y^{2} + w^{3}y^{4})T_{4}^{2}M_{1} + (w^{3}y^{2} + wy^{4})L_{1}L_{2}M_{1} + (w^{5}y^{3} + w^{4}y^{4} + w^{3}y^{5} + w^{2}y^{6})L_{1}T_{4}M_{1} + (w^{7}y + w^{6}y^{2} + wy^{7} + y^{8})L_{2}T_{4}M_{1} + (w^{9}y^{2} + w^{7}y^{4} + w^{3}y^{8} + wy^{10})T_{4}M_{1}.

Proof: All of these relations are detected in a filtered form in spectral sequence (4). Those appearing in the bottom row E_{∞}^{∗,0} are accurate, since there is no room for spillover to lower rows, and have already been listed in formulas (8). The other filtered relations can be expanded to the actual relations by detecting on the two subgroups 2^{2+4} and 2_{−}^{1+4} . We use Theorems 8 and 10 and lemma 28 to compute restrictions, and we use the relations from lemmas 1 and 21, formulas (6), and Theorem 24 to simplify the expressions obtained.

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