The essential ideal in group cohomology does not square to zero

David J. Green

27 February 2002

Abstract

Let G be the Sylow 2-subgroup of the unitary group SU_{3}(4). We find two essential classes in the mod 2 cohomology ring of G whose product is nonzero. In fact, the product is the “last survivor” of Benson–Carlson duality. Recent work of Pakianathan and Yalçın then implies a result about connected graphs with an action of G. Also, there exist essential classes which cannot be written as sums of transfers from proper subgroups.

This phenomenon was first observed on the computer. The argument given here uses the elegant calculation by J. Clark, with minor corrections.

1 Introduction

Let G be a finite group and k a field of characteristic p. A class x in the cohomology ring H^{∗}(G, k) is called essential if its restriction to every proper subgroup is zero. Standard properties of the transfer mean that G has to be a p-group if there are to be nonzero essential classes.

The essential classes in H^{∗}(G, k) form an ideal, called the essential ideal Ess(G). If the group G is elementary abelian, then there is a non-nilpotent essential class: the product of the Bocksteins of all nonzero elements of H^{1}. But in all remaining cases that could be calculated, the pattern that emerged seemed to be that every product of essential classes was zero. This led H. Mui and T. Marx to formulate the following

Essential Conjecture ([8, 6]) Let G be a finite p-group which is not elementary abelian. Then Ess(G)^{2} = 0.

The evidence for the Essential Conjecture consists of a wealth of worked examples, together with one result which is valid for all finite groups: Minh showed in [7] that if x is essential then x^{p} = 0 (assuming G is not elementary abelian).

The Essential Conjecture recently acquired new significance through the work of Pakianathan and Yalçın [9] on G-CW-complexes, as we shall see in Corollary 1.3.

In this paper we disprove the Essential Conjecture. That is, we prove the following:

Theorem 1.1 Let G be a Sylow 2-subgroup of the special unitary group SU_{3}(4). Then Ess(G)^{2} ≠ {0}. To be more precise, there are essential classes in H^{4}(G, 𝔽_{2}) and H^{10}(G, 𝔽_{2}) whose product is nonzero. This nonzero element of H^{14}(G, 𝔽_{2}) is the last survivor in the sense of Benson and Carlson [2]. Moreover there are essential classes in degrees six and eight whose product is the last survivor too.

Theorem 1.1 is proved in Propositions 3.7 and 3.8. Two corollaries can be immediately deduced. The first one was predicted by Minh in [7], who observed that in cases studied to date, the essential ideal had to square to zero, because each essential class was a sum of transfers from proper subgroups. This raises the question of whether essential classes are always sums of transfers.

Corollary 1.2 Let G be a Sylow 2-subgroup of the special unitary group SU_{3}(4). Then Ess(G) is not contained in Tr(G), the ideal in H^{∗}(G, 𝔽_{2}) generated by all transfers from proper subgroups.

Proof. By Frobenius reciprocity, essential classes annihilate transfer classes. So if two essential classes have nonzero product, then neither can lie in Tr(G).

The second corollary is particularly intriguing. It concerns the theory of transformation groups and was predicted by Pakianathan and Yalçın in [9]. Recall from tom Dieck’s book [10, III (3.14)] that if G is elementary abelian and X a finite-dimensional G-CW-complex, then X has a fixed point if and only if the map H_{G}^{∗}(∗) → H_{G}^{∗}(X) is injective. The key assumption here is that G is elementary abelian. Now consider the following corollary:

Corollary 1.3 Let G be a Sylow 2-subgroup of the special unitary group SU_{3}(4) and let X be a connected 1-dimensional G-CW-complex. Then X has a fixed point if and only if the map H_{G}^{∗}(∗) → H_{G}^{∗}(X) is injective.

Proof. This is Theorem 6.2 of [9] (version dated 25 June 2002).

This raises the interesting question of whether there are other groups for which Ess(G)^{r} is nonzero for values of r greater than two. For the groups concerned this would lead by the results of [9] to similar fixed point theorems in higher dimensions. An obvious family of groups to study would be the Sylow 2-subgroups of SU_{3}(2^{n}).

The fact that Ess(G)^{2} is nonzero for the Sylow 2-subgroups of SU_{3}(4) was first observed on the computer [5]. The argument presented here uses J. Clark’s elegant spectral sequence calculation [3], with minor corrections.

2 Clark’s approach to the cohomology calculation

Let G be a Sylow 2-subgroup of the unitary group SU_{3}(4). See Aschbacher’s book [1, §22] for a discussion of unitary groups, and Evens’ book [4] for a general reference on group cohomology. As in Exercise 1 on p. 100 of [1], Clark identifies G with the group of upper triangular matrices ( \begin{matrix}1 & a & b \\ 0 & 1 & a^{4} \\ 0 & 0 & 1 \end{matrix} ) over 𝔽_{16} satisfying b + b^{4} = a^{5}. The centre of G is elementary abelian of rank two, the quotient by the centre is elementary abelian of rank four, and every noncentral element has exponent four. So the centre is the sole maximal elementary abelian subgroup and the cohomology ring over any field of characteristic 2 is Cohen–Macaulay by Duflot’s theorem (Theorem 10.3.1 in Evens’ book [4]).

The cohomology ring of G over the field 𝔽_{16} was calculated by Clark [3] using the Lyndon–Hochschild–Serre spectral sequence of the central extension 1 → Z(G) → G → G/Z(G) → 1. Clark makes elegant use of the Frobenius automorphism and of the action of the normalizer of G.

The normalizer of G in SU_{3}(4) is W G, where W is the cyclic group of order 15 generated by the diagonal matrix T := diag(ζ^{−1}, ζ^{−3}, ζ^{4}) for ζ ∈ 𝔽_{16} is a primitive 15th root of unity. Choose ζ to be a zero of X^{4} + X^{3} + 1, meaning that

ζ^{8} + ζ^{4} + ζ^{2} + ζ = 1 .  (2.1)

The spectral sequence is equivariant for the conjugation action of W, and the action of T on H^{1}(G, 𝔽_{16}) = H^{1}(G/Z(G), 𝔽_{16}) is diagonalisable with eigenvalues ζ, ζ^{2}, ζ^{4}, ζ^{8}. Similarly the action of T on H^{1}(Z(G), 𝔽_{16}) is diagonalisable with eigenvalues ζ^{5}, ζ^{10}. The Frobenius automorphism F: 𝔽_{16} → 𝔽_{16}, F (α) = α^{2} also acts on the spectral sequence, and the spectral sequence for H^{∗}(G, 𝔽_{16}) is just the spectral sequence for H^{∗}(G, 𝔽_{2}) tensored with 𝔽_{16}. Moreover, the actions of T and F commute with each other.

Pick a T -eigenvector a_{1} ∈ H^{1}(G, 𝔽_{16}) with eigenvalue ζ, and set a_{2} = F (a_{1}), a_{4} = F (a_{2}) and a_{8} = F (a_{4}). Then a_{1}, a_{2}, a_{4}, a_{8} is a basis for H^{1}(G, 𝔽_{16}) which is permuted cyclically by F. Here and elsewhere in this paper, a subscript i means that the element concerned is an eigenvector for T with eigenvalue ζ^{i}.

Pick u_{5} ∈ H^{1}(Z(G), 𝔽_{16}) to be an eigenvector of T with eigenvalue ζ^{5}. Then u_{5} ∈ E_{2}^{0,1} and d_{2}(u_{5}) must be a scalar multiple of a_{1}a_{4}, as this is a basis for the eigenspace in E_{2}^{2,0} with eigenvalue ζ^{5}. As G is not elementary abelian, d_{2}(u_{5}) cannot be zero, and so we may choose u_{5} to satisfy d_{2}(u_{5}) = a_{1}a_{4}. Note that u_{5} ∈ H^{1}(Z(G), 𝔽_{4}), since d_{2}(F^{2}(u_{5})) = d_{2}(u_{5}). Setting u_{10} = F (u_{5}), we have that u_{5}, u_{10} is a basis^{1} for H^{1}(Z(G), 𝔽_{16}) which is transposed by F. Moreover,

d_{2}(u_{5}) = a_{1}a_{2} and d_{2}(u_{10}) = a_{2}a_{8} .  (2.2)

An element x ∈ E_{r}^{s,t} is rational (meaning: comes from the spectral sequence for 𝔽_{2}) if and only if F (x) = x. So in particular

H^{1}(G, 𝔽_{2}) = {αa_{1} + α^{2}a_{2} + α^{4}a_{4} + α^{8}a_{8} | α ∈ 𝔽_{16}} .  (2.3)

Notation. Let us denote by N the 𝔽_{2}-linear map N (x) = x + F (x) + F^{2}(x) + F^{3}(x). Then N : H^{n}(G, 𝔽_{16}) → H^{n}(G, 𝔽_{2}) and N : E_{r}^{pq}(𝔽_{16}) → E_{r}^{pq}(𝔽_{2}), the latter being a map of spectral sequences which is H^{∗}(G, 𝔽_{2})-linear but does not in general respect products.

3 Essential classes

We shall now assume that the reader is familiar with Clark’s spectral sequence calculation of H^{∗}(G, 𝔽_{16}), at least as far as the E_{∞}-page. This calculation is performed in Appendix A for the sake of completeness, and because we need an explicit basis for the E_{∞}-page. The appendix also points out some errors in Clark’s relations.

Observe that the ideal of essential classes in H^{∗}(G, 𝔽_{16}) is obtained from the essential ideal in H^{∗}(G, 𝔽_{2}) by extending scalars. So for the Essential Conjecture it is immaterial whether we work over a prime field or over an extension field.

A sufficient (and in fact necessary) condition for a class in H^{∗}(G, 𝔽_{16}) to be essential is that it is divisible by each nonzero element of H^{1}(G, 𝔽_{2}). So a class is essential if it is divisible by N (αa_{1}) for each α ∈ 𝔽_{16}^{∗} (c.f. Eqn. (2.3)).

Lemma 3.1 Every inflation class in H^{4}(G, k) is essential.

Proof. A basis for E_{∞}^{4,0} consists of a_{1}^{4}, a_{2}^{4}, a_{3}^{4}, a_{4}^{4}, a_{1}^{3}a_{2}, a_{2}^{3}a_{4}, a_{4}^{3}a_{8}, a_{8}^{3}a_{1}. Now,

a_{1}^{2}a_{2}N (αa_{1}) = αa_{1}^{3}a_{2} a_{1}^{3}N (αa_{1}) = αa_{1}^{4} + α^{2}a_{1}^{3}a_{2}

by the relations in Equations (A.3) and (A.4). So a_{1}^{3}a_{2} and a_{1}^{4} are both divisible by N (αa_{1}) for each α ∈ 𝔽_{16}^{∗}. Now apply the Frobenius map F.

Notation. We shall say that a class in E_{∞}^{∗∗} is E_{∞}-essential if it is divisible by each nonzero F -stable element of E_{∞}^{1,0}.

1We write u_{5}, u_{10} for Clark’s a_{5}, a_{10}. Apart from this we follow his notation.

Let x ∈ E_{∞}^{pq} be an E_{∞}-essential class. For certain values of p, q we shall show that every class y ∈ H^{∗}(G, 𝔽_{16}) which is represented by x is essential. For q = 0 this is obvious.

Lemma 3.2 Each class in E_{∞}^{6,2} and in E_{∞}^{6,4} is E_{∞}-essential. Hence each element of H^{8}(G, 𝔽_{16}) which is represented by an element of E_{∞}^{6,2} is essential, as is each element of H^{10}(G, 𝔽_{16}) which is represented by an element of E_{∞}^{6,4}.

Proof. The second part follows from the first because E_{∞}^{6,2} and E_{∞}^{6,4} are subspaces of H^{8}(G, 𝔽_{16}) and H^{10}(G, 𝔽_{16}) respectively (see Figure A.1 on p. 7). For the first part, note that the multiplication maps E_{∞}^{2,2} ⊗ E_{∞}^{4,0} → E_{∞}^{6,2} and E_{∞}^{2,4} ⊗ E_{∞}^{4,0} → E_{∞}^{6,4} are surjective. But every element of E_{∞}^{4,0} is E_{∞}-essential, by the proof of Lemma 3.1.

Corollary 3.3 Suppose that x ∈ E_{∞}^{4,4} is E_{∞}-essential. Then every class y ∈ H^{8}(G, 𝔽_{16}) which is represented by x is essential. Similarly, if x ∈ E_{∞}^{6,4} is E_{∞}- essential, then every y ∈ H^{10}(G, k) represented by x is essential.

Proof. Let z ∈ H^{1}(G, 𝔽_{2}) \ {0}. By assumption there is w ∈ E_{∞}^{3,4} such that zw = x. Let w̃ ∈ H^{7}(G, 𝔽_{16}) be a class which is represented by w. Then y − zw̃ lies in the subspace E_{∞}^{6,2} of H^{8}(G, 𝔽_{16}). So by Lemma 3.2 there is v ∈ H^{7}(G, 𝔽_{16}) with y = z(w̃ + v). The second part is proved similarly.

Lemma 3.4 Every element of E_{∞}^{4,4} is E_{∞}-essential. So each element of H^{8}(G, 𝔽_{16}) with trivial restriction to the centre of G is essential.

Proof. The F -orbits of a_{4}^{2}δ_{3} and a_{4}^{2}δ_{7} constitute a basis for E_{∞}^{4,4}. One has

a_{4}δ_{3}N (αa_{1}) = α^{4}a_{4}^{2}δ_{3} + α^{8}a_{4}^{2}δ_{7} and a_{4}δ_{7}N (αa_{1}) = α^{4}a_{4}^{2}δ_{7}

from the definitions in Eqn. (A.8) and the relations in the E_{4}-page. Now use the corollary.

Note that N satisfies the product rule N (A) · N (B) = N (A · N (B)).

Lemma 3.5 Suppose that y ∈ H^{10}(G, 𝔽_{16}) is represented by a class x ∈ E_{∞}^{4,6} of the form x = N (ωa_{4}τ_{6}) with ω ∈ 𝔽_{4}^{∗}. Then y is an essential class.

Proof. The class x is E_{∞}-essential since N (ωα^{11}τ_{6}) · N (αa_{1}) = x + N (ωα^{4}a_{8}τ_{6}) and (ωτ_{10} + ω^{2}τ_{5})N (αa_{1}) = N (ωα^{4}a_{8}τ_{6}). Now use Corollary 3.3.

Observe that (X^{8} +X^{4} +X^{2} +X)(X^{8} +X^{4} +X^{2} +X +1) = X^{16} +X in 𝔽_{2}[X], and that therefore X^{8} + X^{4} + X^{2} + X has eight distinct zeros in 𝔽_{16}. Observe further that these zeros form an 𝔽_{2}-vector subspace.

Lemma 3.6 Suppose that y ∈ H^{6}(G, 𝔽_{16}) is represented by a class x ∈ E_{∞}^{4,2} of the form x = N (λa_{4}^{2}β_{7}) with λ ∈ 𝔽_{16} satisfying N (λ) = 0. Then y is essential.

Proof. Since E_{∞}^{4,2} is a subspace of H^{6}, it suffices to show that x is E_{∞}-essential. Observe that ∏(X^{2} + X + α) = X^{16} + X in 𝔽_{2}[X], where the product is taken over the zeros α of X^{8} + X^{4} + X^{2} + X. So we may pick a μ ∈ 𝔽_{16} satisfying μ^{2} + μ = λ. One then has N (αa_{1}) · N (λα^{11}a_{4}β_{7} + μa_{8}β_{7}) = N (λa_{4}^{2}β_{7}) for all α ∈ 𝔽_{16}^{∗}.

Proposition 3.7 Pick ω ∈ 𝔽_{4}^{∗}. Let η ∈ H^{10}(G, 𝔽_{2}) be the class N (η′) for some η′ ∈ H^{10}(G, 𝔽_{16}) which is represented by ωa_{4}τ_{6} ∈ E_{∞}^{4,6}. Let θ ∈ H^{4}(G, 𝔽_{2}) be the class N (ω^{−1}ζa_{4}^{3}a_{8}). Then η and θ are both essential classes, and ηθ = ξ. Hence Ess(G)^{2} is nonzero.

See §A.3 for the definition of the class ξ. It is the last survivor in the sense of Benson and Carlson [2] for the polynomial generators representing u_{5}^{8}, u_{10}^{8}.

Proof. Lemma 3.1 says that θ is essential, and η is essential by Lemma 3.5. Note that ωa_{4}τ_{6}N (ω^{−1}ζa_{4}^{3}a_{8}) = ζa_{4}^{5}a_{8}β_{7}u_{10}^{4} = ζξ, and so ηθ = N (ζξ). As ξ is F -stable and N (ζ) = 1 by Eqn. (2.1), it follows that N (ζξ) = ξ.

Proposition 3.8 Pick λ ∈ 𝔽_{16}^{∗} to be one of the seven zeros of X^{7} +X^{3} +X +1. Let ϕ ∈ H^{6}(G, 𝔽_{2}) be N (y) for y = λa_{4}^{2}β_{7} in E_{∞}^{4,2} ⊆ H^{6}(G, 𝔽_{16}). Let ψ ∈ H^{8}(G, 𝔽_{2}) be N (z) for some z ∈ H^{8}(G, 𝔽_{16}) which is represented by λ^{−1}ζa_{4}^{2}δ_{7} ∈ E_{∞}^{4,4}. Then ϕ and ψ are essential, and ϕψ = ξ.

Proof. Lemma 3.6 says that ϕ is essential, and ψ is essential by Lemma 3.4. Now λa_{4}^{2}β_{7}N (λ^{−1}ζa_{4}^{3}a_{8}u_{10}^{4}) = ζa_{4}^{4}τ_{6} = ζξ. Therefore N (y)N (z) = N (ζ)ξ = ξ.

Report on the computer calculation The fact that Ess(G)^{2} is nonzero for this group G was first observed using the author’s computer program [5]. As a module over the polynomial algebra k[u_{5}^{8}, u_{10}^{8}], the ideal of essential classes is free with 75 generators. The degrees of these generators are given by the generating function 8t^{4} + 6t^{5} + 3t^{6} + 8t^{7} + 16t^{8} + 7t^{9} + 6t^{10} + 8t^{11} + 8t^{12} + 4t^{13} + t^{14}. The calculations above show that in dimensions 4,5,6,8,10 and 14 there are (at least) as many essential classes as the computer states there to be. The computer states further that Propositions 3.7 and 3.8 account for all nontrivial products of essential classes. In particular, Ess(G)^{3} is zero.

A Clark’s spectral sequence calculation

For Section 3 we need an explicit basis for the E_{∞}-page of the spectral sequence. We shall derive such a basis here. In §A.4 we point out some errors in Clark’s original calculation [3].

From Section 2 we know that the E_{2}-page of the spectral sequence is the polynomial algebra 𝔽_{16}[a_{1}, a_{2}, a_{4}, a_{8}, u_{5}, u_{10}], with the first four generators lying in E_{2}^{1,0} and the last two in E_{2}^{0,1}. From the formula for the differential d_{2} in Eqn. (2.2) one sees that the E_{3}-page is 𝔽_{16}[a_{1}, a_{2}, a_{4}, a_{8}]/(a_{1}a_{4}, a_{2}a_{8}) ⊗ 𝔽_{16}[u_{5}^{2}, u_{10}^{2}]: for a_{1}a_{4}, a_{2}a_{8} is a regular sequence in E_{2}^{∗∗}.

A.1 The E_{4}-page of the spectral sequence

Lemma A.1 The E_{4}-page of the spectral sequence is generated by a_{1}, a_{2}, a_{4}, a_{8} ∈ E_{4}^{1,0}, β_{7}, β_{14}, β_{13}, β_{11} ∈ E_{4}^{2,2} and u_{5}^{4}, u_{10}^{4} ∈ E_{4}^{0,4}. These generators are permuted by F in the obvious way. The classes β_{i} are represented in Z_{5}^{2,2} by

β_{7} = a_{4}a_{8}u_{5}^{2} β_{14} = a_{8}a_{1}u_{10}^{2} β_{13} = a_{1}a_{2}u_{5}^{2} β_{11} = a_{2}a_{4}u_{10}^{2} .  (A.2)

The relations ideal is generated by

\begin{matrix}a_{1}a_{4} a_{2}a_{8}  a_{8}a_{1}^{2} + a_{2}a_{4}^{2} a_{4}a_{8}^{2} + a_{1}a_{2}^{2} \\ a_{1}β_{7} a_{2}β_{7} a_{2}β_{14} a_{4}β_{14} a_{4}β_{13} a_{8}β_{13} a_{8}β_{11} a_{1}β_{11} \\ a_{1}β_{14} + a_{2}β_{13} + a_{4}β_{11} + a_{8}β_{7}  β_{i}^{2} β_{i}β_{j} \end{matrix} .  (A.3)

The following relations also hold:

\begin{matrix}a_{1}^{2}a_{2}^{2} a_{2}^{2}a_{4}^{2} a_{4}^{2}a_{8}^{2} a_{8}^{2}a_{1}^{2}  a_{1}a_{2}^{3} a_{2}a_{4}^{3} a_{4}a_{8}^{3} a_{8}a_{1}^{3} \\ a_{8}^{2}β_{7} + a_{8}a_{1}β_{14} a_{1}^{2}β_{14} + a_{1}a_{2}β_{13} a_{2}^{2}β_{13} + a_{2}a_{4}β_{11} a_{4}^{2}β_{11} + a_{4}a_{8}β_{7} \\ a_{4}a_{8}^{2}β_{7}  a_{8}a_{1}^{2}β_{14}  a_{1}a_{2}^{2}β_{13}  a_{2}a_{4}^{2}β_{11} \end{matrix} . (A.4)

The E_{4}-page is E_{4}^{0∗} ⊗ (E_{4}^{∗0} ⊕ E_{4}^{∗2}). A basis for E_{4}^{∗0} is

\begin{matrix}a_{1}^{r+1}, a_{2}^{r+1}, a_{4}^{r+1}, a_{8}^{r+1}; a_{1}^{r+1}a_{2}, a_{2}^{r+1}a_{4}, a_{4}^{r+1}a_{8}, a_{8}^{r+1}a_{1} for r ≥ 0; \\ 1;  a_{1}a_{2}^{2}, a_{2}a_{4}^{2} \end{matrix} . (A.5)

This basis is permuted by F. One orbit has length one, one has length two, and the remainder have length four. A basis for E_{4}^{∗2} is

\begin{matrix}a_{4}^{r}β_{7}, a_{8}^{r}β_{14}, a_{1}^{r}β_{13}, a_{2}^{r}β_{11} \\ a_{4}^{r+1}a_{8}β_{7}, a_{8}^{r+1}a_{1}β_{14}, a_{1}^{r+1}a_{2}β_{13}, a_{2}^{r+1}a_{4}β_{11} \end{matrix} \} for r ≥ 0; a_{8}β_{7}, a_{2}β_{13}, a_{4}β_{11} .

(A.6)

The orbit sum of a_{8}β_{7} is zero. The rest of the basis is permuted freely by F.

Proof. A basis for E_{3}^{∗0} is: 1; a_{1}^{r+1} and its F -orbit (length four); a_{1}^{r+1}a_{2}^{s+1} and its F -orbit (length four). Here r, s ≥ 0. Note that x^{2} = F ∘ Sq^{r}(x) for x ∈ H^{r}(G, 𝔽_{16}). Hence d_{3}(u_{5}^{2}) = d_{3} Sq^{1}(u_{10}) = Sq^{2} d_{2}u_{10} = a_{1}^{2}a_{8} + a_{2}a_{4}^{2}, and so d_{3} acts as follows:

u_{5}^{2} ↦ a_{8}a_{1}^{2} + a_{2}a_{4}^{2}  a_{1}^{r+1}u_{5}^{2} ↦ a_{8}a_{1}^{r+3}  a_{1}^{r+1}a_{2}^{s+1}u_{5}^{2} ↦ 0
u_{10}^{2} ↦ a_{1}a_{2}^{2} + a_{4}a_{8}^{2}  a_{1}^{r+1}u_{10}^{2} ↦ a_{1}^{r+2}a_{2}^{2}  a_{1}^{r+1}a_{2}^{s+1}u_{10}^{2} ↦ a_{1}^{r+2}a_{2}^{s+3}

Hence E_{4}^{∗0} has the advertised basis, and a basis for Z_{3}^{∗2} consists of the a_{1}^{r}a_{2}^{s}β_{13} for r, s ≥ 0, together with their length four F -orbits. Since a_{i}a_{j}a_{k} = 0 in E_{3}^{3,0} if i, j, k are pairwise distinct, one sees that β_{i}β_{j} = 0 in E_{3}^{4,4} if i ≠ j; and a_{i}β_{j} = 0 in E_{3}^{3,2} if a_{i} does not feature in the definition of β_{j}. Since β_{7}^{2} = d_{3}(a_{4}u_{5}^{4}u_{10}^{2}), one has β_{i}^{2} = 0 in E_{4}^{4,4}. The action of d_{3} on E_{3}^{∗4} is given by

\begin{matrix}u_{5}^{2}u_{10}^{2} ↦ a_{1}β_{14} + a_{2}β_{13} + a_{4}β_{11} + a_{8}β_{7} \\ a_{1}^{r+1}u_{5}^{2}u_{10}^{2} ↦ a_{1}^{r+2}β_{14} + a_{1}^{r+1}a_{2}β_{13} a_{1}^{r+1}a_{2}^{s+1}u_{5}^{2}u_{10}^{2} ↦ a_{1}^{r+1}a_{2}^{s+2}β_{13} \end{matrix} .

So E_{4}^{∗2} has the advertised basis, Z_{3}^{∗4} = u_{5}^{4}Z_{3}^{∗0} + u_{10}^{4}Z_{3}^{∗0}, and the relations ideal is as claimed.

A.2 The E_{∞}-page of the spectral sequence

Proposition A.7 The E_{∞}-page of the spectral sequence is generated by u_{5}^{8}, u_{10}^{8} ∈ E_{∞}^{0,8}, a_{1}, a_{2}, a_{4}, a_{8} ∈ E_{∞}^{1,0}, β_{7}, β_{14}, β_{13}, β_{11} ∈ E_{∞}^{2,2}, δ_{3}, δ_{6}, δ_{12}, δ_{9} and δ_{7}, δ_{14}, δ_{13}, δ_{11} in E_{∞}^{2,4}, τ_{5}, τ_{10} and τ_{3}, τ_{6}, τ_{12}, τ_{9} in E_{∞}^{3,6}, and χ_{5}, χ_{10} ∈ E_{∞}^{3,8}. These generators are permuted by F in the obvious way. Their representatives in Z_{5}^{∗∗} are as follows:

\begin{aligned}δ_{3} & = a_{4}^{2}u_{10}^{4} & δ_{6} & = a_{8}^{2}u_{5}^{4} & δ_{12} & = a_{1}^{2}u_{10}^{4} & δ_{9} & = a_{2}^{2}u_{5}^{4} \\ δ_{7} & = a_{4}a_{8}u_{10}^{4} & δ_{14} & = a_{8}a_{1}u_{5}^{4} & δ_{13} & = a_{1}a_{2}u_{10}^{4} & δ_{11} & = a_{2}a_{4}u_{5}^{4} \\ τ_{3} & = a_{2}β_{11}u_{5}^{4} & τ_{6} & = a_{4}β_{7}u_{10}^{4} & τ_{12} & = a_{8}β_{14}u_{5}^{4} & τ_{9} & = a_{1}β_{13}u_{10}^{4} \end{aligned} ; \begin{aligned}τ_{5} & = (a_{8}β_{7} + a_{4}β_{11})u_{5}^{4} & τ_{10} & = (a_{2}β_{13} + a_{4}β_{11})u_{10}^{4} \\ χ_{5} & = a_{1}a_{2}^{2}u_{5}^{4}u_{10}^{4} & χ_{10} & = a_{2}a_{4}^{2}u_{5}^{4}u_{10}^{4} \end{aligned} .  (A.8)

The E_{∞}-page is E_{∞}^{0∗} ⊗ (E_{∞}^{∗0} ⊕ E_{∞}^{∗2} ⊕ E_{∞}^{∗4} ⊕ E_{∞}^{∗6} ⊕ 𝔽_{16}{χ_{5}, χ_{10}}). A basis for E_{∞}^{∗0} is

\begin{matrix}a_{1}^{r+1}, a_{2}^{r+1}, a_{4}^{r+1}, a_{8}^{r+1}; a_{1}^{r+1}a_{2}, a_{2}^{r+1}a_{4}, a_{4}^{r+1}a_{8}, a_{8}^{r+1}a_{1} for r ≤ 3; \\ 1; a_{1}a_{2}^{2}, a_{2}a_{4}^{2}; a_{1}^{5}, a_{2}^{5} \end{matrix} . (A.9)

This basis is permuted by F. One orbit has length one, two have length two, and the remainder have length four. A basis for E_{∞}^{∗2} is

\begin{matrix}a_{4}^{r}β_{7}, a_{8}^{r}β_{14}, a_{1}^{r}β_{13}, a_{2}^{r}β_{11} & for r ≤ 4 \\ a_{4}^{r+1}a_{8}β_{7}, a_{8}^{r+1}a_{1}β_{14}, a_{1}^{r+1}a_{2}β_{13}, a_{2}^{r+1}a_{4}β_{11} & for r ≤ 2 \end{matrix} a_{8}β_{7}, a_{2}β_{13}, a_{4}β_{11} .

(A.10) q

8  2

7

6  6 8 8 8 4 1

5

4  8 12 8 7 4

3

2  4 7 8 8 8

1

0 1 4 8 10 8 6
0 1 2 3 4 5 6 7 8 p
Figure A.1: The dimension of the E_{∞}-page modulo (u_{5}^{8}, u_{10}^{8})

The orbit sum of a_{8}β_{7} is zero. The rest of the basis is permuted freely by F. A basis for E_{∞}^{∗4} is

\begin{matrix}a_{4}^{r}δ_{7}, a_{8}^{r}δ_{14}, a_{1}^{r}δ_{13}, a_{2}^{r}δ_{11} for r ≤ 4  a_{4}^{r}δ_{3}, a_{8}^{r}δ_{6}, a_{1}^{r}δ_{12}, a_{2}^{r}δ_{9} for r ≤ 2 \\ a_{2}δ_{3}, a_{4}δ_{6}  a_{8}δ_{7}, a_{1}δ_{14}  a_{4}^{3}δ_{3}, a_{8}^{3}δ_{6}, a_{2}^{3}δ_{9} \end{matrix} (A.11)

Apart from the last three terms, this basis is permuted by F. Two orbits have length two, the rest length four. The orbit sum of a_{4}^{3}δ_{3} is zero. A basis for E_{∞}^{∗6} is

\begin{matrix}a_{2}^{r}τ_{3}, a_{4}^{r}τ_{6}, a_{8}^{r}τ_{12}, a_{1}^{r}τ_{9} & for r ≤ 3 \\ a_{2}^{r}a_{4}τ_{3}, a_{4}^{r}a_{8}τ_{6}, a_{8}^{r}a_{1}τ_{12}, a_{1}^{r}a_{2}τ_{9} & for r ≤ 3 \end{matrix} τ_{5}, τ_{10}; ξ ,  (A.12)

where the class ξ = a_{2}^{4}a_{4}τ_{3} is F -stable, and the remaining basis elements are permuted in the obvious way.

Summing up, the dimension of E_{∞}^{pq}/(u_{5}^{8}, u_{10}^{8}) may be read off from Figure A.1 (an empty entry means dimension zero).

The relations ideal for the E_{∞}-page can be recovered from the relations in the E_{4}-page (Equations (A.3) and (A.4)), together with the relations a_{1}^{5} + a_{4}^{5}, a_{2}^{5} + a_{8}^{5} and a_{4}^{3}δ_{3} + a_{8}^{3}δ_{6} + a_{1}^{3}δ_{12} + a_{2}^{3}δ_{9}: these three expressions are the images under d_{5} of u_{5}^{4}, u_{10}^{4} and u_{5}^{4}u_{10}^{4} respectively.

Proof. The β_{i} are permanent cycles for degree reasons, and the classes u_{5}^{4}, u_{10}^{4} are transgressive. So the E_{5}-page coincides with the E_{4}-page. We shall show that the E_{6}-page takes the form claimed for the E_{∞}-page. Then as E_{6}^{r+6,0} = E_{6}^{r+7,2} = 0 for all r ≥ 0, one deduces that there are no further differentials and that E_{6}^{∗∗} = E_{∞}^{∗∗}.

Arguing as for d_{3}, one has d_{5}(u_{5}^{4}) = a_{1}^{5} + a_{4}^{5}. So d_{5} : E_{5}^{∗4} → E_{5}^{∗0} acts thus:

u_{5}^{4} ↦ a_{4}^{5} + a_{1}^{5}  a_{1}^{r+1}a_{2}u_{5}^{4} ↦ a_{1}^{r+6}a_{2}  a_{1}a_{2}^{2}u_{5}^{4} ↦ 0
u_{10}^{4} ↦ a_{8}^{5} + a_{2}^{5}  a_{1}^{r+1}a_{2}u_{10}^{4} ↦ 0  a_{1}a_{2}^{2}u_{10}^{4} ↦ 0
a_{1}^{r+1}u_{5}^{4} ↦ a_{1}^{r+6}  a_{1}u_{10}^{4} ↦ a_{8}^{5}a_{1}  a_{1}^{r+2}u_{10}^{4} ↦ 0

Hence E_{6}^{∗0} has the advertised basis, and a basis for Z_{5}^{∗4} consists of the F -orbits of a_{2}δ_{3}, a_{8}δ_{7} (both length two) and the F -orbits of a_{4}^{r}δ_{3}, a_{4}^{r}δ_{7} (both length four) for r ≥ 0. The map d_{5} : E_{5}^{∗8} → E_{5}^{∗4} is given by

u_{5}^{4}u_{10}^{4} ↦ a_{4}^{3}δ_{3} + a_{8}^{3}δ_{6} + a_{1}^{3}δ_{12} + a_{2}^{3}δ_{9} ; \begin{matrix}a_{4}u_{5}^{4}u_{10}^{4} & ↦ a_{4}^{4}δ_{3} + a_{2}^{4}δ_{11} & a_{4}^{r+2}u_{5}^{4}u_{10}^{4} & ↦ a_{4}^{r+5}δ_{3} \\ a_{1}a_{2}^{2}u_{5}^{4}u_{10}^{4} & ↦ 0 & a_{4}^{r+1}a_{8}u_{5}^{4}u_{10}^{4} & ↦ a_{4}^{r+5}δ_{7} \end{matrix} .

This yields the desired basis for E_{6}^{∗4}, and shows that χ_{5}, χ_{10} are a basis for a complement of u_{5}^{8}E_{6}^{∗0} + u_{10}^{8}E_{6}^{∗0} in E_{6}^{∗8}.

The map d_{5} : E_{5}^{∗6} → E_{5}^{∗2} is given by

a_{4}^{r}β_{7}u_{5}^{4} ↦ a_{4}^{r+5}β_{7}  β_{7}u_{10}^{4} ↦ a_{8}^{4}a_{1}β_{14}  a_{4}^{r+1}β_{7}u_{10}^{4} ↦ 0
a_{4}^{r+1}a_{8}β_{7}u_{5}^{4} ↦ a_{4}^{r+6}a_{8}β_{7}  a_{4}^{r+1}a_{8}β_{7}u_{10}^{4} ↦ 0
a_{8}β_{7}u_{5}^{4} ↦ a_{4}^{5}a_{8}β_{7}  a_{8}β_{7}u_{10}^{4} ↦ a_{8}^{5}a_{1}β_{14}

(For the last statement, note that a_{8}^{6}β_{7} = a_{8}^{5}a_{1}β_{14} by Eqn. (A.4).) Hence E_{6}^{∗2} has the advertised basis, and a basis for Z_{5}^{∗6} consists of the F -orbits of τ_{5} (length two) and of a_{2}^{r}τ_{3}, a_{2}^{r}a_{4}τ_{3} (both length four) for r ≥ 0. The map d_{5} : E_{5}^{∗,10} → E_{5}^{∗6} is given by

β_{7}u_{5}^{4}u_{10}^{4} ↦ a_{4}^{4}τ_{6} + a_{8}^{3}a_{1}τ_{12}  a_{4}^{r+1}β_{7}u_{5}^{4}u_{10}^{4} ↦ a_{4}^{r+5}τ_{6}
a_{4}^{r+1}a_{8}β_{7}u_{5}^{4}u_{10}^{4} ↦ a_{4}^{r+5}a_{8}τ_{6}  a_{8}β_{7}u_{5}^{4}u_{10}^{4} ↦ a_{4}^{4}a_{8}τ_{6} + a_{8}^{4}a_{1}τ_{12}

So E_{6}^{∗6} has the advertised basis, and E_{6}^{∗,10} = u_{5}^{8}E_{6}^{∗2} ⊕ u_{10}^{8}E_{6}^{∗2}.

A.3 The last survivor

Let k be a field of characteristic two. Since H^{∗}(G, k) is a Cohen–Macaulay ring (as noted on p. 2), one sees from Figure A.1 that its Poincaré series P (t) is

\frac{1+4t+8t^{2}+10t^{3}+12t^{4}+13t^{5}+16t^{6}+20t^{7}+16t^{8}+13t^{9}+12t^{10}+10t^{11}+8t^{12}+4t^{13}+t^{14}}{(1−t^{8})^{2}} . (A.13)

Then P (t) satisfies the functional equation P (1/t) = t^{2}P (t), as required by Theorem 1.1 of Benson and Carlson’s paper [2]. This theorem states additionally that H^{∗}(G, k)/(u_{5}^{8}, u_{10}^{8}) satisfies Poincaré duality^{2} in formal dimension 14, the dual pairing being induced by the cup product H^{r}(G, k) ⊗ H^{14−r}(G, k) → H^{14}(G, k).

Notation. Denote by ξ ∈ H^{14}(G, 𝔽_{2}) the unique nonzero F -stable element of E_{∞}^{8,6} ⊆ H^{14}(G, 𝔽_{16}). This class is the last survivor of [2, §7]. By Theorem 1.3 of that paper, it is a transfer from Z(G).

This class ξ is represented by a_{4}^{4}τ_{6} = a_{4}^{5}a_{8}β_{7}u_{10}^{4}, and by any term in its F -orbit.

A.4 Errors in Clark’s calculation

Unfortunately some of the relations in the E_{∞}-page are incorrect on pp. 1424–5 of Clark’s paper [3]. Specifically there are problems with the linear dependencies among the terms δ_{i}a_{j} in E_{∞}^{3,4} and among the terms τ_{i}a_{j} in E_{∞}^{4,6}. The correct relations in E_{∞}^{3,4} are that the following twenty expressions are zero:

\begin{matrix}a_{1}δ_{7} a_{2}δ_{7} a_{2}δ_{14} a_{4}δ_{14} a_{4}δ_{13} a_{8}δ_{13} a_{8}δ_{11} a_{1}δ_{11} \\ a_{1}δ_{3} a_{2}δ_{6} a_{4}δ_{12} a_{8}δ_{9} \\ a_{8}δ_{12} + a_{2}δ_{3} a_{1}δ_{9} + a_{4}δ_{6}  a_{1}δ_{14} + a_{4}δ_{11} a_{2}δ_{13} + a_{8}δ_{7} \\ a_{8}δ_{3} + a_{4}δ_{7} a_{1}δ_{6} + a_{8}δ_{14} a_{2}δ_{12} + a_{1}δ_{13} a_{4}δ_{9} + a_{2}δ_{11} \end{matrix} .  (A.14)

The correct relations in E_{∞}^{4,6} are that the following sixteen expressions are zero:

\begin{matrix}a_{1}τ_{3} a_{2}τ_{6} a_{4}τ_{12} a_{8}τ_{9}  a_{8}τ_{3} a_{1}τ_{6} a_{2}τ_{12} a_{4}τ_{9} \\ a_{1}τ_{5} a_{2}τ_{10} a_{4}τ_{5} a_{8}τ_{10} \\ a_{2}τ_{5} + a_{4}τ_{3} a_{4}τ_{10} + a_{8}τ_{6} a_{8}τ_{5} + a_{1}τ_{12} a_{1}τ_{10} + a_{2}τ_{9} \end{matrix} .  (A.15)

2If k does not contain 𝔽_{4}, then u_{5}^{8}, u_{10}^{8} are not defined over k. Consider instead the parameter system u_{5}^{8} + u_{10}^{8}, ωu_{5}^{8} + ω^{2}u_{10}^{8} for ω ∈ 𝔽_{4} \ 𝔽_{2}. This is defined over 𝔽_{2}.

These relations may be verified using the definitions of the δ_{i}, τ_{j} (Eqn. (A.8)) and the relations in the E_{4}-page (Equations (A.3) and (A.4)). There can be no more such relations, for one sees from Figure A.1 that dim E_{∞}^{3,4} = 12 and dim E_{∞}^{4,6} = 8.

By contrast, Clark states in Figure 1 on p. 1424 of [3] that E_{∞}^{3,4} has dimension ten. This leads to an incorrect coefficient of t^{7} in the Poincaré series at the top of p. 1426: he has 18, whereas the correct value is 20 (see Eqn. (A.13)).

I have not investigated the ungrading problem for the corrected relations; in this paper we just need the E_{∞}-page.

Acknowledgements The author thanks J. F. Carlson, who first drew this group to his attention. The computer calculations mentioned here were performed on a machine belonging to Carlson.

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